Introduction to Complex Analytic Geometry 303487619X, 9783034876193

Table of contents :
Preface to the Polish Edition
Preface to the English Edition
Contents
PRELIMINARIES
CHAPTER A. ALGEBRA
§1. Rings, fields, modules, ideals, vector spaces
§2. Polynomials
§3. Polynomial mappings
§4. Symmetric polynomials. Discriminant
§5. Extensions of fields
§6. Factorial rings
§7. Primitive element theorem
§8. Extensions of rings
§9. Noetherian rings
§10. Local rings
§11. Localization
§12. Krull's dimension
§13. Modules of syzygies and homological dimension
§14. The depth of a module
§15. Regular rings
CHAPTER B. TOPOLOGY
§1. Some topological properties of sets and families of sets
§2. Open, closed, and proper mappings
§3. Local homeomorphisms and coverings
§4. Germs of sets and functions
§5. The topology of a finite dimensional vector space (over C or R)
§6. The topology of the Grassmann space
CHAPTER C. COMPLEX ANALYSIS
§1. Holomorphic mappings
§2. The Weierstrass preparation theorem
§3. Complex manifolds
§4. The rank theorem. Submersions
COMPLEX ANALYTIC GEOMETRY
CHAPTER I. RINGS OF GERMS OF HOLOMORPHIC FUNCTIONS
§1. Elementary properties. Noether and local properties. Regularity
§2. Unique factorization property
§3. The Preparation Theorem in Thom-Martinet version
CHAPTER II. ANALYTIC SETS,ANALYTIC GERMS, AND THEIR IDEALS
§1. Dimension
§2. Thin sets
§3. Analytic sets and germs
§4. Ideals of germs and the loci of ideals. Decomposition into simple germs
§5. Principal germs
§6. One-dimensional germs. The Puiseux theorem
CHAPTER III. FUNDAMENTAL LEMMAS
§1. Lemmas on quasi-covers
§2. Regular and k-normal ideals and germs
§3. Rückert's descriptive lemma
§4. Hilbert's Nullstellensatz and other consequences (concerning dimension, regularity, and k-normality)
CHAPTER IV. GEOMETRY OF ANALYTIC SETS
§1. Normal triples
§2. Regular and singular points. Decomposition into simple components
§3. Some properties of analytic germs and sets
§4. The ring of an analytic germ. Zariski's dimension
§5. The maximum principle
§6. The Remmert-Stein removable singularity theorem
§7. Regular separation
§8. Analytically constructible sets
CHAPTER V. HOLOMORPHIC MAPPINGS
§1. Some properties of holomorphic mappings of manifolds
§2. The multiplicity theorem. Rouche's theorem
§3. Holomorphic mappings of analytic sets
§4. Analytic spaces
§5. Remmert's proper mapping theorem
§6. Remmert's open mapping theorem
§7. Finite holomorphic mappings
§8. c-holomorphic mappings
CHAPTER VI. NORMALIZATION
§1. The Cartan and Oka coherence theorems
§2. Normal spaces. Universal denominators
§3. Normal points of analytic spaces
§4. Normalization
CHAPTER VII.
ANALYTICITY AND ALGEBRAICITY
§1. Algebraic sets and their ideals
§2. The projective space as a manifold
§3. The projective closure of a vector space
§4. Grassmann manifolds
§5. Blowings-up
§6. Algebraic sets in projective spaces. Chow's theorem
§7. The Rudin and Sadullaev theorems
§8. Constructible sets. The Chevalley theorem
§9. Rückert's lemma for algebraic sets
§10. Hilbert's Nullstellensatz for polynomials
§11. Further properties of algebraic sets. Principal varieties. Degree
§12. The ring of an algebraic subset of a vector space
§13. Bézout's theorem. Biholomorphic mappings of projective spaces
§14. Meromorphic functions and rational functions
§15. Ideals of 𝒪ₙ with polynomial generators
§l6. Serre's algebraic graph theorem. Zariski's analytic normality theorem
§17. Algebraic spaces
§18. Biholomorphic mappings of factorial subsets in projective spaces
§19. The Andreotti-Salmon theorem
§20. Chow's theorem on biholomorphic mappings of Grassmann manifolds
REFERENCES
[10a]
[21]
[35]
NOTATION INDEX
SUBJECT INDEX
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Citation preview

Stanislaw Lojasiewicz

Introduction to Complex Analytic Geometry Translated from the Polish by Maciej Klimek

1991

Springer Basel AG

Author's addrcss: Dr. Stanistaw Lojasicwicz Jagicllonian University Dcpartmcnt of Mathcmatics ul. Rcymonta 4 PL-30-05'1 Cracow (Poland) Originally published as: Wst (p·1)· (q .1)-1 E K). If A is an integral domain of characteristic zero, then (8 i- 0, x i- 0) =* 8X i- 0 for 8 E Z and x E A. Note also that if a ring A contains (as a subring) a field of' characteristic zero, s

----->

(lOa)

An element of A is said to be reducible if it is a product of two non-invertible elements.

(11) Or, equivalently, if s . 1

=1= 0 for all s E Z \

{OJ.

8

A.l.IS

A. Algebra

then, for each proper ideal I, the ring A/lis of characteristic zero. (This is because, in such a case, the elements s . I, where s E Z \ {O}, are invertible and hence they do not belong to I.) 15. If A is a subring of a field L, then by the field of fractions of the ring A in the field L we mean the subfield generated by A. It is equal to the set of elements of the form xy-I where x E A and yEA \ 0, and is isomorphic to the field of fractions of the ring A via the A-isomorphism xy-I ---+ x/yo In particular, when A is a subring of an integral domain B, the field offractions of A will be identified with the subfield (generated by A) of the field of fractions of the ring B. If A, AI are subrings of the fields L, V, respectively, then every isomorphism between the rings A and AI can be extended to an isomorphism between their fields of fractions in Land L I , respectively. Let f: L ---+ V be a field isomorphism. If I{ is the field of fractions of the subring A in L, then f(I{) is the field of fractions of the subring f(A) in V. If I{ is a subfield of the field Land I{I = f(I{), then dimK L = dimK' LI. 15a. Let A be a ring, and let 5 be the complement of the set of zero divisors in A. There exists a unique (up to an A-isomorphism) extension of the ring A in which all elements of 5 are invertible and every element of which is of the form xy-I, where x E A, y E 5. The proofs of existence and uniqueness as well as the construction of the canonical extension with these properties are exactly the same as in the case of the field of fractions of an integral domain. The only difference is that the "denominators" are taken from 5, i.e., one begins the construction with the set A x 5 (see [1], Chapter 1, §4; or [4], Chapter 3). The extension is called the ring of fractions of the ring A. Obviously, when A is an integral domain, one gets the field of fractions of A (since, in this case, 5 = A \ 0).

Let A be a subring of a ring AI, and let R, RI be the rings of fractions of the rings A and AI, respectively. If all elements of the ring A which are not zero divisors in A are not zero divisors in AI, then - after identification via the well-defined monomorphism R 3 x/y ---+ x/y E RI (12) - R is a subring of the ring RI. Note also that if R I , ... , Rk are the rings of fractions of the rings AI, ... , A k , respectively, then the ring of fractions R of the ring Al x ... X Ak (12) Then the diagram A

AI

----+

RI

! R

commutes.

!

§l.

9

Rings, fields, modules, ideals, vector spaces

is isomorphic to the ring RI x ... ... X Ak )-isomorphism

X

Rk via the well-defined natural (AI x

If B is an extension of the ring A, then a denominator (in A) of an element b E B is an element a E A which is not a zero divisor and is such that ab E A. When each element of B has a denominator in A, the ring B can be identified - in a natural way - with.a sub ring of the ring of fractions R of the ring A. The identification is obtained via the (well-defined) monomorphism B 3 b - ; ab/a E R (where a is a denominator of the element b). An element a E A which is not a zero divisor and is such that aB C A (i.e., a common denominator for all elements of B) is called a universal denominator of the extension B (or - more precisely - of the ring B over the ring A). Obviously, if B is an integral domain and has a universal denominator over A, then the fields of fractions of both rings coincide.

16. A finite module M over A is said to be free if it has a basis, i.e., a sequence of linearly independent generators Xl, ... ,X n . If that is the case, M ~ An because An 3 (tl, ... ,tn) - ; I:~tixi E!vI is an isomorphism. For any n, the module An is free and it has the canonical basis el = (1,0, ... ,0), ... , en = (0, ... ,0,1). Hence a module (which is finite over A) is free precisely when it is isomorphic with one of the modules An. Clearly, a (finite dimensional) vector space is always a free module. Obviously, if M is a free module, there exists a unique homomorphism of the module M into another one that takes arbitrarily given values on an arbitrarily given basis of M. It follows that if f: L - ; M is an epimorphism (of a module L), there exists a monomorphism g: M - ; L such that fog = id M . With every module homomorphism f: An - ; Am one can associate its matrix Cf' This is the unique (m x n)-matrix [Cij] E A~ (with entries Cij E A) such that n

(Yl,···,Ym) = f(xl,""x n ) ~Y;

=

2::c;jXj for i

=

l, ... ,m.

j=l

If n = m, the mapping f -; C f is an isomorphism between the noncommutative ring of all endomorphisms of the module Anand the noncommutative ring A~ of all (n x n )-matrices with entries from A. We define det f as det Ct.

10

A.1.16a

A. Algebra

C E A~ there exists a matrix (det C)I, where I is the identity matrix. If Xl, ... ,X n are elements of a module over A and 2:7=1 CijX j = 0 for i = 1, ... , n, then (det Cij )xs = 0 for s = 1, ... ,n. If det C is invertible, then the matrix C is invertible. If det f is invertible, then f is an automorphism. CRAMER'S THEOREM. For each matrix

D E A~ such that CD

=

DC

=

(See [3], Chapter XIII, §4; and 2.2a below.) Let M be a module over a ring A. We have the following MATHER-NAKAYAMA LEMMA. Let C be a subring (not necessarily with identity, e.g. an ideal) of the ring A, and let ~ E A. If M is finite over

C(13), then "lM

= 0 for

some "l E ~m

+ c~m-l + ... + C.

Indeed, let aI, ... , am be generators of Mover C. Then m

~ai =

L lijaj ,

j=l

where lij E C, and in view of Cramer's theorem (det(~8ij -lij))a s 1, ... ,m.

= 0,

s =

16a. Let B be an extension of a ring A. We say that B is fiat over A if every solution (Xl, ... , Xn) E Bn of any linear equation 2:~ Xiai = 0 with coefficients aI, ... , an E AT is a linear combination (with coefficients from B) of solutions of this equation that belong to An (see [l1a]' Chapter I, 2.11).

An equivalent reformulation of this definition is as follows. Let an Ahomomorphism be any module homomorphism h: Bn ------> BT such that h(An) c A". Note that A-homomorphisms are exactly the mappings of the form Bn:3 (Xl, ... ,Xn) ------> 2:~ Xiai E BT, whereo.l,'" ,an E AT. Thus B is flat over A if and only if the kernel of every A-homomorphism Bn ------> BT is a submodule generated by a subset of An. If B is flat over A, then the ring of polynomials B[X] is flat over the ring of polynomials A[X]. This can be verified easily by expressing linear equations involving polynomials in terms of linear equations involving the coefficients of the polynomials instead (see 2.1 below).

17. Let M be a vector space. The codimension of a subspace L of AI, i.e., the dimension of the space MIL, is denoted by codim Lor codimML. Let Hand L be subspaces. If M = L + H, then codim L S dimH. If L n H = 0, then codim L 2': dimH. Hence, if ]v! = L + H is a direct sum, (13) i.e., M =

2: CXi with some Xl, ... ,Xk E M

(as for rings with identity).

11

§l. Rings, fields, modules, ideals, vector spaces

then codim L = dimH. In particular, codimL = dimM - dimL, provided that M is finite dimensional. (Since the natural linear mapping H ~ M / L is surjective and injective, respectively, in the previously considered cases.) Therefore L. has finite dimension if and only if M = L + H for a finite dimensional subspace H. If L C H, then codim L ;:::: codim H, and hence H has a finite codimension if L does. Moreover, if codim L = codim H < 00, then L = H. (For the natural epimorphism MIL ~ MIH becomes an isomorphism in such a case.) 18. Let M be an n-dimensional vector space over a field K. We say that the subspaces L 1 , ... , Lk C M intersect transversally (or that they are in

n:

L; =I- 0 for all systems of affine subspaces L~, ... ,L~ general position) if which are parallel to L 1 , ••• ,L k , respectively. This happens precisely when

n k

(7)

codim

k

Li =

L: codim Li

Indeed, let us take the natural linear mapping 'Pi: M ~ M / L i . The fact that subspaces L; intersect transversally means exactly that the mapping

(with the kernel n~ Li) is surjective. Since dimM = 2:~ codimL;, this equivalent to (7). Note that the inequality

n k

codim

IS

k

Li S;

L: codim L;

is always true. Observe also that the subspaces Li intersect transversally if and only if in some linear coordinate system ('P: M - + J(n) they are of the form

where

2:

h, ... , In are disjoint subsets of {I, ... , n}. In fact, the condition is sufficient, as we have codim

n

Ti =

#

U Ii

2:

=

2: #Ii

(n

=

codimTi. Conversely, if the Li 's intersect transversally, then the sum L; = Li).1 (15) is direct because dim L; = codim Li = codimLi = dim L;. Hence we

2:

n

2:

2:

(14) It means that 'P( Li) = Ti. (15) For any subspace L C !'vI of dimension k, L.1 = {'P EM: 'PL = O} is a subspace of dimension n - k of the dual space M*. See B. 6.4.

12

A.1.l9

A. Algebra

can choose a basis 'f'I, ... , 'f'n of the dual space M· , such that 'f'v, v E I;, generate Lt6 1, ... ,n) and the I;'s are disjoint. Then 'f' ('f'I, ... ,'f'n): M --+ J(n ) is an isomorphism and L; {'f'v 0 for v E I;} 'f'-l(T;).

(i

1

=

:S

=

=

=

e

=

It follows that if Ll' ... , Lk intersect transversally, then so do L"'l"'" L",., where < ... < a. :S k.

al

Two subspaces L, HeM intersect transversally if and only if L

+H =

M. (Since, in this case, the condition (T) reduces to the equality dime L + H) = n.) If, in addition, we know that dim L + dim H = n, the condition of transversality of the intersection of Land H can be expressed as L

n H = O.

If HI, . .. ,Hn are hyperplanes, then they intersect transversally if and only if n~ Hi = O. In the same way one defines transversality of the intersection of a family of affine spaces L l , ... , Lk C M by requiring that n~ L;

i= 0 and condition (T)

is fulfilled (17) .

19. Notice that if c.p: M ---4 X is an epimorphism of vector spaces, then for any isomorphism of the form X = (c.p, 7r): M ---4 X X Y, where 7r: M ---4 Y = ker c.p is a projection, and for the natural projection p : X X Y ---4 X, the diagram

M

X

XxY

---4

commutes. Then

x (c.p -1 (E)) = E x Y for E eX. 20. Let M and L be vector subspaces over a field K, and let f: lvIk ---4 L be a k-linear alternating mapping. If a = (aI, ... ,an) is a permutation of the set {I, ... , k}, then

where COt denotes the sign of the permutation (18). If 1 have the identity n

f(I: 1

n

aljXj, ... ,

I: 1

akjXj)

L

~

k

~

n, then we

(det a;)..j )f(X)..l'··· ,X)...)

1::;)..1 < ... n. This is the case because, if k > n, it suffices to substitute X n +1 Xk 0 in the k-th Newton identity in Z[X I , ... , Xk).

= ... =

=

Therefore, if a ring A contains a field of characteristic zero, the theorem on symmetric polynomials is true when 0'1, ... ,0' n are replaced by the S ). polynomials 81, ... , 8 n

e

3. Since the polynomial fL (2). Conversely, it follows from the condition (2) that P and pI have a common irreducible divisor, say G. Thus P = GH and pi = GF for some F, HE A[X]. Hence G' H = G(F - H'). Therefore G divides Hand G 2 divides P. Let K be the field offractions for the ring A, and let L be an extension of K such that P can be factorized in L[X] into linear factors: P = (X - (1) ... (X - (n). Then condition (3) means that PI((r) = 0 for some r. Hence (2) yields (3). Conversely, note that there is a greatest common divisor of the polynomials P and pI in K[X] which is primitive in A[X]; it is their common divisor in A[X] and their greatest common divisor in L[XJ. Therefore it is of positive degree if condition (3) is fulfilled. PROOF.

§7. Primitive element theorem A primitive element of an extension L of a field K is an element ( E L which is algebraic over K and such that L = K((). Notice that in such a case, for any c E K \ 0 the element c( is then primitive. THE PRIMITIVE ELEMENT THEOREM. Every finite extension L of a field K of characteristic zero has a primitive element. Moreover, if Zl, ... ,Zr are infinite subsets of the field K and L = K( 1)1, ... ,1)r), then there exists a primitive element r

PROOF. First of all, it is enough to show the existence of a primitive element of the form "'11 + C2"'12 + ... + CrTJr for arbitrarily given infinite sets 9 Z2,"" Zr C K, where Ci E Zi ) . Secondly, it is sufficient to prove this

e

e

9

)

(with

For then we can take C E Zl \ 0 and the primitive element (' Ci E c- 1 Zi) and define ( C(l

=

= 1)1 +

C21)2

+ ... + Cr 1)r

30

A.S.l

A. Algebra

statement for r = 2 (the general case would follow by induction). Let L = K( 0:, f3), where 0:, f3 E L are algebraic elements over K. Let f, 9 E K[X] be their minimal polynomials, and let L' be an extension of the field L such that f, 9 can be factorized in L' [X] into linear factors: f = (X - 0:1) ... (X - O:m) and 9 = (X - f3d ... (X - f3n), where 0:1 = 0: and f31 = f3. Since L' is of characteristic zero and f, 9 are irreducible in K[X], the o:;'s, as well as the f3/s, are mutually distinct. Now, there is e E Z2 such that O:i + efJj -I- 0: + efJ for i = 1, ... m and j = 2, ... ,n (because for each such pair i, j the equality could be true for at most one such c). Put ( = 0: + ef3. Clearly, K(() c K( 0:, f3). In order to prove the opposite inclusion, consider the polynomial h = f(( - eX) E K(()[X]. Obviously, h(fJ) = 0 and h(fJj) -I- 0 for j 2: 2 (because ( - cfJj -I- O:i for i = 1, ... , m, for such j's). Therefore among the factors X - fJj, only X - fJ is a common divisor of the polynomials 9 and h in L'[X]. It follows that it is their greatest common divisor in L'[X]. Let d denote their greatest common divisor in K( ()[X]. Then d is also their greatest common divisor in L'[X], and hence d = a(X - fJ), where a E L' \ O. This implies that a, afJ E K( (). Thus f3 E K( () and 0: = ( - efJ E K( (). Hence

K( 0:, fJ) C K( (). An extension L of a field K is said to be algebraic (over K) if each element of L is algebraic over J(. COROLLARY (40). Let L be an algebraic extension of a field K of characteristic zero. Then L is a finite extension of K if and only if ihe degree nx of x over K is a bounded function on L. Then its maximum n is equal to dimJ( L, and {x E L: nx = n} is the set of all primitive elements of the extension.

In fact, if the extension is finite, then nx .:::; dimJ( L. Now, assume that CXl and nx = n. For any z E L we have K(x) C K(x, z) = K(w) for some wE L, and dimJ( K(w) .:::; n = dimJ( K(x) (see 5.3), which implies that K(x) = K(x,z), i.e., z E K(x). Therefore L = K(x). n

= supnx


(I is isolated for a non-zero principal ideal).

This gives us the following characterization of noetherian rings that are factorial. A noetherian integral domain A is factorial if and only if each of its prime ideals of height 1 is principal.

Indeed, the condition is sufficient. To see this, let. x be an irreducible element, and let I be an isolated ideal for Ax. Then h(I) 1 and hence I Az for some element z

=

=

56

A.13.1

A. Algebra

which must be non-invertible. So x az for some element a which must be invertible. Thus I Ax, i.e., the element x is prime. Therefore (see 9.5) the ring A is factorial. Now assume that A is factorial, and let I be one of its prime ideals of height 1. Then, for some c :p 0, the ideal I is isolated for Ac, and thus the element c must be non-invertible. Hence c Xl ... Xk, where Xi are prime (see 6.1). But then x. E I for some 8 (since c E 1), and so the ideal I is isolated for Ax 8' Therefore I Ax 8'

=

=

=

3. Let A be a noetherian local ring with the maximal ideal m. We define the (Krull) dimension of the ring A by the formula

dimA

= max{k:

IoCj. ... Cj. h}

= h(m)

,

where Iv are prime ideals (94). Thus we have THEOREM 1". The dimension of a noetherian local ring A is equal to the minimum of the numbers of generators of the defining ideals of A, z.e., dim A

= min{g(I): I

is defining} .

Every system of generators of a defining ideal which realizes this minImUm, i.e., every system of (dim A) elements which generates a defining ideal, is called a system of parameters of the ring A. Thus every noetherian local ring has a system of parameters.

PROPOSITION . If an element x

E

m is not a zero divisor, then

dimAIAx = dim A - 1. PROOF. Set s = dim AI Ax. There are prime ideals II, ... , Is+ 1 of the ring A such that Ax c II Cj. ... Cj. Is+I = m. But the ideal II is not isolated for 0, as it contains x (see 9.3). Thus there exists a prime ideal IoCj. II. Consequently, dimA ~ s + 1. On the other hand, there is a defining ideal J :) Ax, such that J lAx is generated by some elements Xl, ... , xs , where Xl, ... , Xs E J. But then J must be generated by Xl, ... ,x s , x, and thus dim A ::; s + 1. Observe also that if h : A rings, then dim B ::; dim A.

----4

B is an epimorphism of noetherian local

We also show the following

LEMMA. Let len c A be ideals, and suppose that n is prime. Then the dimension of the rings

(see 11.2) is equal to max{k: Ie Ioc;. ...

c;. hen} ,

where Ii are prime ideals.

(94) Therefore dim AI I ::::: dim A for every proper ideal I (see 1.10-11).

57

§13. Modules of syzygies Indeed, the prime ideals of

An/l

is a prime ideal satisfying the inclusion

are precisely those of the form

l cJ

J/l,

where

len

(see 1.10-1.11 and 11.2), which is equivalent

to Ie J (95) .

§13. Modules of syzygies and homological dimension 1. Let A be a local ring with the maximal ideal m.

Let M be a finite module over A. For any minimal system of generators a = (al, ... , an) for the module M we define a submodule of the module An by syzaM

= {(t 1 , ... , t n ) EAn:

t

tiai

= O}

I

This submodule is equal to the kernel of the epimorphism

A n :7 (tI, ... ,tn)

n --t

Ltiai EM.

Therefore M ~ An /syzallif. Each of the modules syzaM is called a module of syzygies for the module M. Notice that syza1l1 c mAn = m X ... X m (since 2: tiai = 0 ==? t I, ... ,tn E m in view of the minimalit.y of the system a). If M and N are finite modules over A that are isomorphic, then any two of their modules of syzygies are isomorphic. In particular, all modules of syzygies of the module M are mutually isomorphic. PROPOSITION.

Let J. associated with /.+1. For otherwise (see 1.11 and 9.3) there would exist an

=

(lOB) See [17b], III. 1.1, prop. 3.

=

65

§14. The depth of a module

element t E J, which is not a zero divisor in A/ 1.+ 1. Then, by (#) (in view of the proposition 2), tl, ... , t s , t would be an (A/I)-sequence, and hence, by (#) (see 9.3), we would have t ¢ J,. Thus the sequence J o , . .. ,Jk is defined. Finally, t 8 +1 E J s + 1 \ J, (because ts+l E 1,+1, and by (#)). Therefore (see 12.3): prof A/ I

~

dim A/ J

~

dim A/ I

for each J associated with I .

In particular (taking 1== 0), we have prof A

~

dim A .

The ring A is said to be a Cohen-Macaulay ring if prof A == dim A. Thus, if A/I is CohenMacaulay, then all the rings A/ J, where J is associated with I, have the same dimension (equal to dim A/I). Hence we have COROLLARY 1. If A/I is Cohen-Macaulay, then every ideal associated with I is isolated for I. (For if

h¥ J 2

are prime, then dimA/JI

> dimA/J2 ; see

1.10-11.)

Proposition 4 from nO 2 and the proposition from 12.3 imply COROLLARY 2. If A is Cohen-Macaulay and t Em is not a zero divisor, then A/At .s also Cohen-Macaulay (109). PROPOSITION 7. The ring A is Cohen-Macaulay if and only if one of its systems of parameters is an A -sequence. Then each of its systems of parameters is a maximal A -sequence. PROOF. In view of (*), the condition is sufficient. Thus it is enough to prove that if A is Cohen-Macaulay, then each of its systems of parameters is an A-sequence. Set n == dim A. If n == 0, then, by theorem 1" from 12.3, the zero ideal is defining, and so m k == O. It follows that m contains only zero divisors, i.e., prof A == O. Now, let n > 0, and suppose the statement is true for n - 1. Let A be Cohen-Macaulay, and let t 1 , . .. ,tn be a system of parameters. Set A == A/At n . It is easy to check (see 1.4 and 10.3 and 5) that [1, ... , f n - 1 generate a defining ideal in A. We claim that tn is not a zero divisor. If it were, then (see 9.3) tn would belong to an ideal associated with 0 and we would have dim A ;::: dim AI J ;::: n (see 12.3), in view of proposition 6. This is impossible because dim A ~ n - 1. Therefore, by corollary 2 and the proposition from 12.3, A is CohenMacaulay of dimension n - 1. Consequently, ll, ... , In-l is a system of parameters of A (

09

)

This implies a more general statement: If A is Cohen-Macaulay and t l

is an A-sequence, then A/ L::~ Ati is also Cohen-Macaulay. For if B

==

, ... ,ik

E m

A/ L::~-I Ati

is Cohen-Macaulay, then (since we have Blk == L::~ At;! L::~-I Aii) so is A/ L::~ Ati ~ B/ Blk , because lk is not a zero divisor in B. It follows that the localization An of a CohenMacaulay ring A to a prime ideal n is Cohen-Macaulay. In fact, in view of proposition 1, there exists an A-sequence which cannot be extended (by an element of n). Then (see

[1, ... ,

footnote (0 3 )), [k E fi is an A-sequence and the ring A/I, where I == L::~ Ati, is Cohen-Macaulay. Hence, by corollary 1, all ideals associated with I are isolated. One of them must contain n (for otherwise - see 1.11 and 9.3 - the ideal n would contain a non-zero divisor in A/I, contrary to the inextendability of the sequence ii). Hence it must coincide with n, and so, by theorem l' from 12.2, we get prof An;::: k ;::: h(n) == dim An (see 11.2). Therefore, in view of (*), it follows that An is a Cohen-Macaulay ring.

66

A.15

A. Algebra

and hence an A-sequence. By lemma 1 and proposition 5 it follows (in view of proposition 2) that tl, ... ,tn is an A-sequence.

§15. Regular rings Let A be a noetherian local ring with maximal ideal m. We always have edim A = g(m) ;:: dimA (110) . The ring A is said to be regular if edim A = g( m) = dim A, i.e., m has n generators, where n = dim A. Then, obviously, each system of n generators is minimal and vice versa. Thus all elements of such a system belong to m \ m 2 and each element of m \ m 2 belongs to some such system. Also, dim A > 0 {:=? m i- 0 {:=? m \ m 2 i- 0. A regular ring is zero dimensional if and only if it is a field. THEOREM 1. Let A be a regular ring of dimension n > 0, and let x = (Xl, ... , xn) be a system of generators of the ideal m. Then, for each k > 0,

the collection {xP}!p!=k is a minimal system of generators for the ideal mk. The theorem yields yet another characterization of regular rings (which can be easily

=

seen to be equivalent to the theorem). Namely, Ao = A/m, Al = m/m 2 , .•. , Ak mk /m k +}, ... are vector spaces over I< = A/m, and we have the natural bilinear mappings

Ak x AI 3 (t, u) --+ tu E Ak+l. The direct sum G(A) = defined by

EB:

O:=xi)(LYi) == LXjYj

(where

Xv,Yv

EB:

Av with the multiplication

E A,,) is a graded ring, which is said

to be associated with A. (A ring B is said to be graded if it possesses a decomposition

B

==

B" into the direct sum of subgroups of the additive group of B, such that

BkBI C Bk+l - see [1], Chapter 1, §5; or [4], Chapter 10.) We have a natural gradation in the ring of polynomials I< [X} , ... ,XnJ

==

EB:

H v , where H" is the vector space that

consists of forms of degree v. Let n = dim A.

The ring A is regular precisely when

G(A) where

~

~

I< [Xl , ... , Xn],

denotes an isomorphism of graded rings, i. e., when there is an isomorphism = Av.

r.p: I< [Xl , ... ,XnJ--+ G(A) such that r.p(Hv)

For if such an isomorphism exists, we may assume it is a I< -isomorphism (by replacing it by r.p 0 t/J-l, where t/J is the automorphism of the ring I H is proper, where GeM is open and dense (and I(G) C H), then we must have G = 1-1 (H). (Indeed, in this case G must be closed in 1-1 (H) - see below.) (8) A family of sets R is said to be centred (or to have the finite intersection property) if El n ... n Ek # 0 for any finite collection El , ... , Ek E R.

77

§3. Local homeomorphisms and coverings

G :J f-1(b) such that the family of compact sets f-1(6.) \ G, where 6. varies over all compact neighbourhoods of the point b, would be centred; hence its intersection f- l (b) \ G would be non-empty. Note also that if the restriction of a continuous mapping f: M ~ N to a locally closed set E is proper, then the set E must be closed. (Otherwise, by taking a point a E E \ E and compact neighbourhood V of the point f( a), the set (JE)-l(V) = f-I(V) n E would not be closed.)

§3. Local homeomorphisms and coverings Let M and N be Hausdorff spaces. 1. Let the mapping f: M ~ N be a local homeomorphism (i.e., for each a E M the restriction of f to a neighbourhood of the point a is a homeomorphism onto a neighbourhood of f(a)).

Then f is continuous and open. Hence it is enough to consider only open neighbourhoods in the definition of a local homeomorphism. If the mapping

f

is bijective, it is a homeomorphism.

Suppose now that continuous mappings hi: E ~ M satisfy f(hi(z)) = Z on the set E eN, for i = 1,2. If hJ (c) = h2(C) for some c E E, then hJ = h2 in a neighbourhood in E of the point c. Thus, if the set E is connected, then either hI = h2 in E or hI i:- h2 in E. (Due to the fact that the set {hl(z) = h2(Z)} is open and closed in E.)

If M is connected and there is a continuous mapping h: N M such that f(h(z)) = z in N, then f must be a homeomorphism. LEMMA.

Indeed, it is enough fact that heN) = {x: heN). The restriction fu homeomorphism onto the

~

to show that h is surjective. Now, owing to the h(J(x)) = x}, the set heN) is closed. Let a E of f to a neighbourhood U of the point a is a neighbourhood feU) of the point c E f(a); since

h(c) = a = (JU)-l(c), we have h = (JU)-l in a neighbourhood V C feU) of the point c, and then h(V)

= (Ju) -\V)

is a neighbourhood of the point a

which is contained in heN). Thus the set heN) is open. Hence heN) = M. 2. A mapping f: M ~ N is said to be a covering if every point of the space N has an open neighbourhood V such that f-l(V) is the union of

78

B.3.2

B. Topology

some open sets U, which are pairwise disjoint and for which the mappings fv, : U, ---t V are homeomorphisms e). Every covering is a local homeomorphism (and hence an open and continuous mapping). Let f: M ---t N be a covering. For any point c E N we can find - as in the definition - a neighbourhood V and sets U, such that there are exactly #f-l(z) sets U, for every z E V; hence the number #f- 1 (z) depends only on c. It is called the multiplicity of the covering f at the point c. When regarded as a function of c, it is locally constant. In the case when it is constant in N, its only value p is said to be the multiplicity of the covering f and we also say that the covering f is Jrsheeted. In particular, this is the case (for some p) when the space N is connected. Every I-sheeted covering is a homeomorphism. We say that a covering f is finite when its multiplicity is finite at each point of the space N. If f: M

---t

N is a covering, then for E C N the restriction

ft-l(E) :

f- (E) 1

---t E is also a covering. The Cartesian product of coverings is a covering. Clearly, the composition of a covering with a homeomorphism (in any order) is a covering.

If M and N are locally connected, then the restriction of a (finite) covering f: M ---t N to any closed open subset of the space M (in particular, to a connected component of the space M) is a (finite) covering eO) . 1. If M and N are locally compact, then for any mapping N we have the following equivalence:

PROPOSITION

f: M

---t

(f is a finite covering)

{=}

(f is a local homeomorphism and a proper mapping) . PROOF. Suppose f is a finite covering. Then any point of the space N has a neighbourhood whose inverse image is compact. (It is enough to take a compact neighbourhood V such that f- 1 (V) = ](1 U ... U ](s, where the IC's are homeomorphic to V.) Thus the mapping f is proper.

Assume now that f is a local homeomorphism and a proper mapping. Let c EN. Then the set f-l(C) is compact and discrete (for if a E f-l(c), fv is injective on a neighbourhood U of a). Hence f-l(c) is finite, say f-I(C) = {aI, ... ,a r }. Take disjoint neighbourhoods UI , ... , Ur ofthe points aI, ... ,a r , (9) In such a case, the same property is displayed by any open neighbourhood V' C V together with the sets U: U, n f-l(V').

=

eO) Compare with the previous footnote.

79

§3. Local homeomorphisms and coverings

respectively, such that fUi is a homeomorphism and f(Ui) is a neighbourhood of the point c for i = 1, ... , r. Take a compact neighbourhood V* of c. The set T = f(J-l (V*) \ (U l U ... U Ur )) is compact and does not contain the point c. Therefore there is an open neighbourhood U of the point c which is disjoint from T and such that V C V* n f(U i ). Then f-l(V) c Ul U ... U Ur (since x E f-l(V)\(U l U .. . UUr ) would imply that f(x) E TnV). Thus f-l(V) =

n

U{ U ... U U;, where U: = Ui n f-l(V) are open and disjoint. Moreover, the mappings fu;: U: --+ fCUD = f(Ui) n V = V are homeomorphisms. Hence the mapping

f

is a finite covering.

PROPOSITION 2. If M I- 0 is connected and N is homeomorphic to R n, then every covering f: M --+ N is one-sheeted, i. e., is a homeomorphism. PROOF. Without loss of generality we may assume that N = R n. By the lemma, it is enough to show the existence of a continuous mapping h : N --+ M satisfying f(h(z)) = z in N. Since the multiplicity of f is constant,

f must be surjective, and so there is an a E M such that f(a) = 0. Let T be the class of all M -valued continuous mappings 9 such that their domains are open and star-shaped with respect to 0, g(O) = a, and f(g(x)) = x in the domain of g. The class T is non-empty, as it contains (Ju) -1 for a suitably chosen neighbourhood U of the point a. Now h = U{g: gET} is a mapping and it belongs to T. (Any two of the mappings in T coincide on the intersection of their domains, since the latter is connected and contains 0.) It suffices to show that the domain H of h is equal to N. Suppose that it is not true and H* N. Then there exists a point c E N \ H such that [0, c) C H. Let us take an open convex neighbourhood V of the point c for which f- 1 (V) is the union of open disjoint sets U, and fu, : U, --+ V are homeomorphisms 1) . Consider b E [0, c) n V and an open convex set W such

e

that [0, b] eWe H. We have h(b) E UK for some

K-,

and hence h(b)

where h = (JuJ -1; hence f(h(z)) = z in V. Thus h = h in W

= h(b),

n V, and

therefore hw U h is a continuous mapping on the set W U V :J [0, c] and its restriction to an open convex set W o, such that [0, c] C Wo C Wo U V, belongs to T. In conclusion, Wo CHand hence c E H, which is a contradiction.

80

B.4.1

B. Topology

§4. Germs of sets and functions Let S, T, U be topological spaces. 1. Let a E S. By the germs of sets at the point a (in the space S) we mean the equivalence classes of the equivalence relation

"E' n V = E" n V for some neighbourhood V of the point a " in the set of all subsets of the space S. The equivalence class of a set E is called the germ of E at the point a and is denoted by Ea. The relation of inclusion, the operation of taking the finite union or intersection of sets, the difference of sets or the complement of a set, together with their elementary properties from the algebra of sets, carryover in a natural way to germs at a (12). In this new context, the role of the empty set and the whole space are played, respectively, by the empty germ 0, i.e., the germ of the empty set and the full germ, i.e., the germ of the whole space (the representatives of the latter germ are precisely neighbourhoods of the point a). The above operations are well-defined by the formulae Ea U Fa = (E U F)a, Ea n Fa = (E n F)a, etc. The inclusion of germs A C B is defined by the condition that ii C B for some representatives ii, B of those germs. Thus, for sets E, Fe S, the inclusion Ea C Fa means that En V c F n V for some neighbourhood V of the point a. Note also that if a E E, then every neighbourhood in E of the point a is a representative of the germ Ea and every representative of the germ Ea contains a neighbourhood in E of the point a. For a germ A at the point a and a set E C S we write ACE when A C Ea, i.e., when A is the germ of a subset of the set E. We define also En A = An E = A n Ea. The germ of the set {a} at the point a will be denoted simply by a. We will also write that a E A if A is the germ of a set containing the point a.

If a E Sand bET, then the formula Ea x Fb = (E x F)(a,b), where E C Sand F C T defines the Cartesian product of germs. (The same works for an arbitrary finite number of germs.) Let h be a homeomorphism of a neighbourhood of the point a onto a neighbourhood of the point b = h(a) E T. The image of a germ at a (under h) is well-defined, as a germ at b, by the formula h(Ea) h(E)b, where E C S. Obviously, h(A U B) = h(A) U h(B), h(A n B) == h(A) n h(B), (12) In what follows, all the necessary properties can be checked easily.

81

§4. Germs of sets and functions

and A C B {==:} h(A) C h(B) for germs A, B at a. If h(A) = C, then h- 1 (C) = A. For any homeomorphism g of a neighbourhood of the point b onto a neighbourhood of the point g(b) E U, we have g(h(A)) for every germ A at a.

= (g 0 h)(A)

If a ESC T is a subspace and E C S, then the germ of the set E in the space S at the point a is identified with the germ of this set in the space T at a. (This identification is compatible with inclusion, finite union, and finite intersections of germs, but not with taking complements.) 2. Let A be a germ of a set at the point a E S, and let X be an arbitrary set. By the germs of functions from A to X we mean the equivalence classes with respect to the following equivalence relation

"F' = F" on a representative of the germ A" in the set of all X -valued functions defined on representatives of the germ A. The equivalence class of such a function F is called the germ of F on A and denoted by FA. Also, for any function F whose domain contains A, the germ FA = (FA)A is well-defined, where A is a representative of the germ A. We will use the symbol f: A ----4 X to express the fact that f is the germ on A of an X -valued function. In the case when X is a ring (or a module over a ring R), the above relation agrees with the multiplication and addition of functions (13) (or multiplication of a function by elements from R, respectively). As a result, all those operations can be defined on the set of the germs on A of X -valued functions (14) and they furnish this set with a structure of a ring (or a module over R, respectively). The restriction of the germ f: A

----4

X to a germ C C A is well-defined

by the formula fe = ie, where F is a representative of the germ f. Then, for any germ DeC, we have fD = (fe)D. If a E A, then the value of the germ f at a is well-defined by the formula f(a) = Jea). Let B be the germ of a set at the point bET. The germ f: A ----4 T is said to be the germ of a mapping of the germ A into the germ B. We write f: A ----4 B if every representative of the germ B contains the range of some representative of the germ f. In the case when a E A, this happens only if f is the germ of a mapping from a representative of the germ A to a representative (13) Note that f

+ g and

fg are defined on the intersection of the domains of f and g.

(14) The operations are well-defined by the formulae Fa (FG)a (or (Fa = ((F)a, respectively).

+ Ga =

(G

+ G)a

and (FaGa) =

82

B.4.3

B. Topology

of the germ B which is continuous at a and assumes the value b at a (15). Then, if B' is the germ of a set at the point b, we have f: A ---t B' precisely when f has a representative whose range is contained in some representative of the germ B'. In particular, f: A ---t B' if B' ::J B. Sometimes we will use the symbol f B' instead of f, to indicate that the germ f is treated as f: A ---t B'. If h is the germ of an X -valued function defined on the germ C of a set in U and g: B ---t C, then the composition hog: B ---t X is well-defined by the formula hog = (hog)B, where h,g are representatives of of the germs h, g. We have 9 0 f: A ---t C and the composition of germs is associative: ho(go1) = (hog)of. When g: B ---t DeC, then hDog = hog. We denote by eA the germ of the identity mapping on a representative of A. Clearly, eA: A ---t A' if A C A'. We put e1' = (eA)A' and {!A = (eA)s •. Obviously, f 0 eA = f and f = eB 0 f. If C C A, then (eA)c = ee and, for any germ h of a function on A, he = h 0 ee. Therefore (g 0 1)e = go fe for any germ 9 of a function on B. The diagonal product of germs gl, ... , gk of functions on B with values in Y1 , ... , Y n , respectively, is well-defined by the formula (gl,'" ,gk) = (gl,' .. ,gk)B, where gl, ... ,gk are representatives ofthese germs with a common domain. We have (gl," . ,gk) 0 f = (gl 0 f,···, gk 01) and, in particular, (gl, ... ,gk)D = ((9t)D, ... ,(gk)D) if DeB. Similarly, we can define the Cartesian product of germs of functions. Assume now that a E A or A = 0, and b E B or B = 0. The germ of a homeomorphism of the germ A onto the germ B is the germ f on A, of a homeomorphism] of a representative of the germ A onto a representative of the germ B taking the value b at a, if A and B are non-empty. (Naturally, f: A ---t B.) In this case, the inverse germ f- 1 is well-defined as the germ on B of the homeomorphism ]-1 and we have f- 1 0 f = eA, f 0 f- 1 = eB, and

(I-I) -1

= f. If C C A, the image of the germ C is well-defined by the

formulaf(C) = ](C). IfD = f(C), then C = f- 1 (D), andmoreover,ifC:3 a or C i- 0, then the restriction fe: C ---t D is the germ of a homeomorphism of C onto D. The composition of the germs of homeomorphisms (of A onto B and of B onto C) is the germ of a homeomorphism (of A onto C). If f: A ---t Band g: B ---t C are the germs of continuous mappings such that go f = eA and fog = eB, then f and 9 are mutually inverse germs of homeomorphisms (of A onto B and of B onto A) (16). (15) Note that, in general, the ranges of representatives of the germ f (even if the domains are sufficiently small) need not represent the same germ of a set at the point b. To see this, take the germ at 0 of the mapping R2 3 (x, y) ---+ (x, xy) E R2. (16) Indeed, assuming that a E A, b E B, and taking representatives

j: A ---+

T, g :

83

§4. Germs of sets and functions

If G is a homeomorphism that maps a neighbourhood of the point bET onto a neighbourhood of a point c in U and C is the germ of a set at the point c, then GB, where B = G-1(C) is the germ of a set at b, is the germ of a homeomorphism of the germ B onto the germ C. A substitution into the germ h of a function on C is defined by hoG = hoG B. Thus, if F is a homeomorphism of a neighbourhood of the point a in S onto a neighbourhood of the point b in T, then we have the associative law (h 0 G) 0 F = h 0 (G 0 F). If X is a ring (resp., m9dule) and f: A ---. B, then we have a ring (resp., module) homeomorphism 'T/ ---. 'T/ 0 f of the ring (resp., module) of germs of X-valued functions on B into the ring (resp., module) of germs of X-valued functions on A. It becomes an isomorphism if f is the germ of a homeomorphism of the germ A onto the germ B. Similarly, the mappings 'T/ ---. 'T/e, where a E A, are homomorphisms of the ring (resp., module) of germs of X -valued functions on A into the ring (resp., module) of germs of X-valued functions on C or into the ring (resp., module) X. 3. In the case of the full germ A = Sa, the germ FA is said to be the germ of the function F at the point a and is denoted by Fa. Hence it is the equivalence class of the function F defined in a neighbourhood of a, with respect to the equivalence relation (in the set of all the X -valued functions that are defined in a neighbourhood of the point a) given by: "F' = F" in a neighbourhood of the point a".

Note that if a E E C S, then the set of germs of functions on A = Ea can be identified with the set of germs of functions at a in the space E via the natural bijection fA ---. fa (where the f are functions on neighbourhoods of the point a in the space E). In the case when X is a ring (resp., module), the bijection becomes a ring (resp., module) isomorphism. 4. Let f: S ---. X and g: S ---. X. Sometimes we will write f =: 9 instead of f = 9 in S (and f =: 9 in E instead of f = 9 in E for any E C S). The symbol f cj. 9 will be used to denote the fact that fx f gx for all xES (i.e., that f =: 9 is not true in any open non-empty subset or, equivalently, that the set {f(x) f g(x)} is dense). Hence, if X is a Hausdorff topological space and f, 9 are continuous, f cj. 9 precisely in the case when the set {f( x) = g( x)} is nowhere dense. If E C S, then f cj. 9 in E will indicate that fE gE·

t

13

---> S for sufficiently small neighbourhoods U, V of the points a, b in A, 13, respectively, we have g(i(x) = x in U and f(g(y) = y in V. But feU) is always a neighbourhood

of the point b in

j(g(W)

gV : V

= W).

--->

13

(because U ::J g(W) for some neighbourhood W C V, and so j(U) ::J

Therefore we may assume that V U are mutually inverse homeomorphisms.

= feu);

then

fu: U

--->

V and

84

B. Topology

B.S.1

§5. The topology of a finite dimensional vector space (over C or R)

1. Let X be an n-dimensional real or complex vector space (i.e., over R or over C). Then X has a (unique) natural topology such that some (and hence each) linear coordinate system on X is a homeomorphism 7) . Every norm on X defines the same natural topology. Thus every two norms are equivalent and X, regarded as a normed space, is complete. Every vector subspace of the space X is closed. If E eX, then

e

(E is compact) {:::::::} (E is closed and bounded), (E is relatively compact) {:::::::} (E is bounded). A set E is said to be bounded if it is contained in a ball with respect to a norm on X. (This condition is independent of the choice of the norm and the centre of the ball.) Finally, the space X is locally compact. The set of all linearly independent sequences (Xl' ... ' Xn) E Xk is open in X k and the set of all sequences (Xl, ... , X r) E xr that generate xr is open in xr. In particular, the set of all bases of the space X is open in xn. In the space X, the closure of a cone is a cone. Any compact set which does not contain the origin generates a closed cone. (For the cone generated by a compact non-empty set E C X \ 0 is 7r({(z,w) E X x E: Z 1\ w = OJ), where the natural projection 7r: X x E --+ X is closed.) If S, T C X are cones, then To C So {:::::::} T C S, and hence To

= So

{:::::::} T C S.

(In other words, a cone is determined by its germ at 0.) 2. Let X, Y, Z, X', Y' be finite dimensional (real or complex) vector spaces. Every linear mapping from the space X to the space Y is continuous. In the case when it is surjective, it is also open. (17) It is, at the same time, the only Hausdorff topology on X under which X is a topological vector space (i.e., the algebraic operations on X are continuous). If X is a vector space over C, then, naturally, it is a vector space over R and both natural topologies coincide.

§s. The topology of a finite dimensional vector space

85

Every polynomial mapping P: X ---T Y is continuous. If P = 0 on a non-empty open set, then P == 0 (18). Therefore the condition P ;j; 0 is the negation of the condition P == O. If P is non-zero, the set {P = O} is closed and nowhere dense, whereas the set {P :j:. O} is open and dense. In particular, the operations

+ and·

are continuous (19).

The set L(X, Y) of all linear mappings on the space X into Y is a finite dimensional vector space and dim L(X, Y) = (dim X)( dim Y). The composition L(X, Y) x L(Y, Z) 3 ('P,?/;) ---T ?/; 0'P E L(X, Z) , the diagonal product

L(X,Y) x L(X,Z) 3 ('P,?/;)

---T

('P,?/;) E L(X,Y x Z),

and the Cartesian product

L(X,Y)

X

L(X',Y') 3 ('P,'P')

---T

'P x 'P' E L(X

X

X',Y

X

Y')

are continuous mappings eO). Note also that the mapping

L(X,Y) x X

3

(J,x)

---T

f(x)

E

L

is continuous (21). The function L(X, Y) 3 'P

---T

rank'P E N is lower semicontinuous.

Indeed, if rank 'Po = r, then 'Po (ad, ... ,'Po (a r ) are linearly independent for some aI, ... ,a r . Thus, for each mapping 'P from some neighbourhood of the mapping 'Po, the elements 'P(ad, ... ,'P(a r ) are also linearly independent, and hence rank 'P :::: r. Accordingly, the sets {'P E L(X, Y): rank'P:::: k}, kEN, are open. In particular, both the set of all monomorphisms in L(X, Y) and the set of all epimorphisms in L(X, Y) are open (22). (18) It is sufficient to prove the statement in the case of a polynomial P: where K = C or K = R. Now, the property holds true when n = 1. Let n > 1 that it is also true in the case of n - 1 variables. It follows that P = 0 in G X non-empty open sets G C K n -1 and If C K. Thus P = 0 in K n - 1 x If, and Kn-1 X K.

K n --+ K, and assume H for some so P = 0 in

(19) On X2 and C x X or R x X, respectively. eO) The same holds for the composition, the diagonal product, and the Cartesian product of a finite number of mappings.

(21) Since the above mappings are linear (the first three) or bilinear (the last one) and hence polynomial.

(22) By taking k

= dim X

and k

= dim Y, respectively, we can obtain these two sets.

86

B. Topology

B.6.1

If dimX = dim Y, then the set of all isomorphisms Lo(X, Y) C L(X, Y) is open and dense. Moreover, the mapping Lo(X, Y) :3 rp ~ rp-1 E Lo(X, Y) is open and dense in the vector space of the n X n-matrices (as C ~ det C is a non-zero polynomial) and the mapping {det C i- O} :3 C - - t C- 1 E {det C 1= O} is continuous. A linear mapping f: X - - t Y is proper precisely in the case when ~ 00 as Ixl ~ 00. (Notice that it would be equivalent to each of these two conditions to require that the inverse image of any bounded set is bounded.)

If(x)1

Let M be a locally compact topological space, and let E C M x Y be a locally closed set. Then the natural projection E ~ M is proper if and only if E is closed and each point of the space M has a neighbourhood U such that the fibres Ex = {y E Y: (x, y) E E}, for x E U, are uniformly bounded, i.e., they are all contained in a common ball (with respect to a norm in Y). 3. Let zn + a1zn-1 + ... + an be a monic polynomial with complex coefficients. Observe that (1, ... , (n is a complete sequence of its roots (23) precisely when aj = 0"]((1, ... , (n) for j = 1, ... , n. Then

(Iail (For if

1(1 > 2r,

~ r,i = 1, ...

,n) =? (I(jl ~ 2r,j = 1, ... ,n).

then

Consequently, the polynomial mapping

is a proper surjection. (Since the inverse image of any bounded set is bounded in view of the fundamental theorem of algebra.) Observe that 0"( Z1 , ... , zn) = o if and only if Z1 = ... = Zn = O. THE THEOREM ON CONTINUITY OF ROOTS. Let (1, ... , (n be a complete sequence of roots of the polynomial zn + a1 zn-1 + ... + an. Then for each 15 > 0 there exists 8 > 0 such thai if ICi - ai I < 8, i = 1, ... , n, then IZj - (j I < 15, j = 1, ... , n, for a (suitably ordered) complete sequence of roots Zl, •.. , Zn of the polynomial zn + CIZ n - 1 + ... + Cn.

87

§6. The topology of the Grassmann space

PROOF. The set E = noUi{Z E en : Izo; - (d 2 c}, where a = (aI, ... , an) varies over all permutations of {I, ... , n}, is symmetric and closed. Furthermore, a-I (a (E») = E. Indeed, if a(z) = a(w) and wEE, then z is a permutation of w (since both z and w are sequences of the roots of the same polynomials), and so z E E. Hence we have a(\E) =

a(\aq(a(E»))

=

a(a- 1 (\a(E»))

=

\a(E)

(24). But

aCE) is closed, be-

cause a, being a proper mapping, is closed. Therefore the set

a(\E) =

a(U n{z E en: o

IZa; -

(d < c})

i

is open. Since it contains the point (al,' .. ,ak) = a( (1, ... ,(n), it must also contain the set {( Cl, ... ,c n ): ICi - ai I < 6} for some 6 > O. Thus, if ICi - ai I < 6, i = 1, ... ,n, then Cl, ..• ,C n is the sequence of the coefficients of a monic polynomial whose roots are Zl, . . . ,Zn, and which is such that Izo; - (il < c, i = 1, ... , n, for some permutation a.

§6. The topology of the Grassmann space Let X be a complex n-dimensional vector space

e

S

).

1. For any kEN, the Grassmann space Gk(X) is the set of all kdimensional subspaces of the space X. In particular, P(X) = G 1 (X), i.e., the set of all lines passing through 0 is called the projective space of dimension n -1 and we let dimP(X) = dimX -1 (26). Clearly, Gk(X) = 0 for k > n.

From now on we will assume that 0 ::; k ::; n. Let Bk(X) denote the set of all linearly independent sequences z E Xk. The set is open in Xk. In particular, B(X) = Bn(X) is the set of all bases of the space X. In the Grassmann space Gk(X) we can introduce the following topology. The open sets are defined to be the sets whose inverse images under the surjection

(24) The symbol \E denotes the complement of the set E.

(25) In the present context, the real case does not differ from the complex one. (26) In Chapter VII we will give P(X) the structure of an (n - 1)-dimensional complex manifold (see VII. 2.1).

88

B.6.2

B. Topology

are open in Bk(X) (27). Then the surjection a is continuous and open. (Indeed, if G C Bk(X), then a - I (a(G)) = UA(G), where

A: Xk E

(Xl' ... '

Xk)

---t

k

k

I

I

(L aljXj, ... , L

akjXj) E Xk , det aij =I- 0,

are isomorphisms and hence also homeomorphisms.) The space Gk(X) is compact. To see this, note first that it is a Hausdorff space. This is so since the diagonal in Gk(X?, i.e., the set {(U, V): U = V}, is closed, as its inverse image under the open surjection a X a (see 2.1) is the set {(X,y) E Bk(X)2: Xl /\ ... /\ Xk /\ Yi = 0, i = 1, ... ,k} , which is closed in Bk(X)2. Secondly, the space Gk(X) is the image under a of the compact set of all orthonormal sequences in Xk (with respect to a fixed Hermitian product in X). The space Gk(X) is connected (see n° 8 below). The topology of the space Gk(X) has a countable basis (furnished by the image under a of any countable basis in Bk(X)). Thus the notions of open sets, closed sets, etc. can be characterized in Gk(X) in terms of the convergence of sequences 8 ) .

e

If Z C X is a vector subspace, then the topology of the space Gk(Z) coincides with that induced by the topology of the space Gk(X). (Indeed, the mapping Gk(Z) '---+ Gk(X) and its inverse are both continuous; see 2.2.) The set Gk(Z) is closed in Gk(X). Moreover, it is nowhere dense, provided that k > 0 and ZCj. X (see 2.2). Note that if G c X is an open set, the set {I E Gk(X): LnG =I- 0} is open in Gk(X) (as it is equal to a({x E Bk(X): Xl E G})). Thus, if Fe X is a closed set, then the set {L E Gk(X): L C F} is closed in Gk(X). 2. In the space Gk(X) x G/(X), where k Un V = O} is open, and the mapping s: {U

e

7)

nV

= O} :3 (U, V)

---t

U

+I

+V

::; n, the set {(U, V)

E Gk+I(X)

Or, equivalently, in Xk.

(28) Note that, in the space Gk(X), a sequence U v converges to U if and only if for some (and hence any) basis x in U there exists a sequence Xv ---+ x, where Xv is a basis in U v for v 1,2, .... This is a consequence of the fact that the image under a of a base of neighbourhoods of an x in Bk(X) is a base of neighbourhoods of a(x) in Gk(X) (see 2.2).

=

89

§6. The topology of the Grassmann space

is continuous. (Indeed, ak x al is a continuous and open surjection, the set (ak X al)-I({U n V = O}) is open, and the mapping so (ak X al) C ak+l is continuous - see 2.2.) 3. We have the following

Let Y be a finite dimensional complex vector space. The mapping B(X) x yn :3 (x, y) --+ fxy E L(X, Y) is continuous, where fxy is given by fxy(Xi) = Yj (i = 1, ... , n). LEMMA.

Indeed, the mapping

where FxU) = (t(XI),"" f(xn)) is linear and hence continuous. Now, if (x,y) E B(X) x yn, then FxUxy) = y, Fx is an isomorphism and we have fxy = Fx-l(y). This yields the continuity of the mapping (x, y) --+ fxy (see 5.2). 4. With every k-dimensional subspace V of the space X one can associate the (n - k)-dimensional subspace V~ = {cp E X*: CPV = O} of the dual space X*. We have the following properties

(VI n ... n Vr)~ = V/

+ ... + V/,

(VI

+ ... + Vr)~

U C V - V~, O~

= X*,

X~

= V/

=0

n ... n Vr~ ,

.

Moreover, if cp: X --+ Y is an isomorphism of vector spaces, then cp*(V~)

= cp-I(V)~

for any subspace V C Y .

We will prove that the bijection

is a homeomorphism. In fact, since Gk(X) is compact, it is enough to show that the mapping

T

is continuous. Set a*

=

a:~k' Let x E Bk(X), Fix

t = (tk+l,"" t n ) E xn-k in such a way that (x, t) E B(X). Then (z, t) E B(X) for z in some neighbourhood ltV of the point x. Now, for z E ltV, the forms f; E X*, i = k + 1, ... ,n, given by f;(zv) = 0, v = I, ... ,k, and f!(t j) = Ojj, j = k + 1, ... , n, constitute a basis of the space a( z)~. Hence a(z)~ = a*Uz), where fz = U;+1, .. · ,I,}). This implies the continuity of

90

B.6.5

B. Topology

the mapping (T 0 O')w, because the mapping W :7 Z ---4 fz is continuous by the lemma. Therefore the composition TOO' is continuous and so is the mapping T (see 2.2). 5. For any subspace U C X of dimension::::; k we define the Schubert cycle: Sk(U)

= Sk(U, X) = {V

E Gk(X): V::) U} .

Clearly (see nO 4),

(#)

= Gn_k(U.l..) T(Gk(V)) = sn-k(v.l..) T(Sk(U))

for any subspace U of dimension::::; k ,

2 k.

for any subspace V of dimension

Accordingly, the Schubert cycle Sk(U) is a closed subset of the space Gk(X), In addition, it is nowhere dense provided that k < nand U :f. 0 (see n° 1). 6. The function

is lower semicontinuous. Indeed (see 2.2), by composing it with O'Pl x ... apr we obtain a restriction of the lower semicontinuous function XP :3 C (Cl,'" ,cp ) ---4 rankg c E N (29), where P = PI + ... + Pr and gc : CP :7 Z (Zl, •.. , zp) ---4 ZiCi E X (see 5.2).

L:i

Therefore the function Gk l (X) x ... X Gkr(X) :7 (VI, . .. , Vr )

... n Vr ) E N is upper semicontinuous (as dim(VI n ... n Vr ) =

--+

X

= =

dim(VI n

n - dim(VI.l..

... + V/)).

+

Consequently, The set {(VI, ... , Vr ) : dim(VI n Gk l (X) x ... X Gk r (X) for all sEN.

... n Vr )

::::;

s} is open in the space

This implies (see A.l.18) that the set of sequences of subspaces (VI," . . . . , Vr ) E G kl (X) X .•. X G kr (X) which intersect transversally is open. Next, the set {(U, V):

U C V} in the space Gp(X) x Gk(X), where

0::::; P ::::; k is closed (since it is equal to {(U, V): dim(U n V) 2 p}). 7. Let C Gk(X) be a closed set. For 0 ::::; P ::::; k, the set U{Gp(V) : V E } is closed in Gp(X). (This is so, because it is the .image of the compact set (Gp(X) x ., Y = L- + f-L, and so there exists an isomorphism Y is said to be C~linear (or R~ linear) if it is linear as a morphism of the complex (resp., real) vector spaces X,Y. An R~linear mapping '{! : X --> Y is C~linear if and only if '{!( ix) = i'{!( x) for all x EX. The space cn, regarded as a vector space over R, is identified with· (R2 = R 2n. When an n-dimensional complex vector space is regarded as a vector space over R, it is 2n-dimensional.

t

In what follows, we will be assuming that all vector spaces (over C or R) are finite-dimensional. 2. Let f be a function defined in a neighbourhood of a point a E C, with values in a complex vector space Y. Then the complex derivative

1· f(a + z) - f(a) a = Jnl "--'---'---'--'---'f '() Z~O

e)

Z

It is enough to restrict multiplication to R x X. On the other hand, if X is a vector space over R, it can be endowed with the structure of a complex vector space (whose underlying real structure coincides with the original one), provided that X is even dimensional. This can be achieved by introducing an endomorphism x ----t ix, such that i( ix) = -x, which would play the role of multiplication by i. Consequently, the multiplication by complex numbers would be given by the formula (a + f3i)x = ax + ,B(ix).

§l.

99

Holomorphic mappings

exists if and only if f is differentiable at a e) and the differential daf is C-linear e). In this case, daf(z) = J'(a)z. (This is because the condition that f (( a + z) - f( a)) / z --+ C as z --+ 0 is equivalent to the condition that f(a

+ z) -

f(z)

= cz + o(z)

as z --+ 0.) Moreover, ¥x-(a)

~~ (a) = iJ'(a) (because daf(x, y) = J'(a)x

+ iJ'(a)y).

= J'(a)

and

Now let f be a function defined on a neighbourhood of a point a E cn and with values in a complex vector space Y. If f is differentiable at the point a (4), then the differential daf is C-linear if and only if the partial derivatives ;!.(a),v

= 1, ... ,n, exist.

Then

Indeed, we have daf(z) = L~=l Vdaf(z), where vdaf denotes the differential of the mapping (--+ f( ... ,aV-l,(,a V+1,"') at a v and a = (a}, ... ,a n ). Therefore each of the above conditions is equivalent to C-linearity of the differentials v daf, v = 1, ... , n. 3. We say that the series L cp (or more precisely, LNn cp), where cp E C and pENn, is convergent and its sum is equal to C E C (this is also written as C = L cp ) if, for each E > 0, there exists a finite subset Zo C Nn such

that for each finite subset Z ::::l Zo we have I Lz Cp - ci < E (5). Then the series Llpl2:: k C p (6) is also convergent and L cp = Llpl 0 there exists a finite subset Zo C N" such that subset Z disjoint from Zo.

(6) That is, the series

L

c~, where c~ = 0 for

Ipl < k,

IL z

and c~

cp I


U', 'ljJ: v', where u' ,V' are open sets of complex vector spaces x', yl, respectively, such that

v->

'ljJofUO O. Assume that On-I is regular and (n -I)-dimensional. By Hadamard's lemma (see C. 1.10), the kernel of the epimorphism On '3 I - + I(ZI,"" zn-I, 0) E On-I (see nO 2) is equal to the ideal Onzn. Therefore On/(Onzn) ~ On-I, which implies that the ring On is regular of dimension n (see A. 15 lemma 1, and the proposition in A. 12.3).

Note that the germs II, ... , In in the ideal rna generate rna if and only if their differentials dalI, ... ,daln are linearly independent. Indeed, it suffices to consider the case of On. The mapping m '3 g - + dol E L = (cn)* is a linear epimorphism whose kernel is m 2 (see n° 7). Thus m/m2 '3 I - + dol E L is an isomorphism, where I denotes the equivalence class of the germ I E m. But the germs J; generate m if and

Ii

only if generate the linear space m/m 2 (see A. 10.4) or, equivalently, if the differentials doli generate L. The last condition means that the differentials are linearly independent (13). Observe that the rank of an ideal I C rna of Oa (see A. 10.4) can be expressed by means of the generators g}, ... ,gk of I as follows:

(13) Here is an alternative proof. If /j generate

m=

n

m,

a aij E On. Thus bij = " . aij (0) aa fi (0), and so det i!.li. a ZJ: (0) DJ Z1

#

then Zi

= Lj aij /j

O. Conversely, if det

for some

¥'- # 0, z}

af· ( 0) = bik for some Cij E C. Therefore then "6 j Cij ~

Zi

=L

Cij ( L k J

which implies that

See also II. 4.2.

mC

I:j

~~: (Oh) = L

Cij/j(Z)

+ o(lzl) ,

J

On!j

+ m2 . Finally, in

view of the Nakayama lemma,

§2. Unique factorization property

145

For, in the case of On, the image of (1 +m2)jm2 by the isomorphism mjm 2 - - t L induced by the epimorphism m ::l ! - - t do! E L is equal to the image of 1 by this epimorphism, i.e., to the subspace 2: Cdog i whose dimension is rank o(g1,'" ,gk)'

§2. Unique factorization property 1. A germ from Qn is said to be distinguished of degree k if it is the germ of a distinguished polynomial in Zn of degree k; in other words, if it is monic, of degree k and with all coefficients (but the leading one) belonging to m n -1 (see C. 2.2). It is, clearly, regular of order k. Furthermore, it is non-invertible in On (or, equivalently, in Qn) precisely when k > 0 (15).

The classical version of the Weierstrass preparation theorem (see C. 2.4) can now be stated as follows:

A ny germ from On which is regular of order k is associated in On with a unique distinguished germ. Moreover, the degree of the distinguished germ is k. Consequently, distinguished germs that are associated in On must coincide. Note also that if a germ 9 E Qn is distinguished and h E On, then gh E Qn =? hE Qn. Indeed, we have gh = qg + r, where q, r E Qn and the degree of r is less than that of 9 (see A. 2.4). But the degree of the germ 9 is equal to the order of g. Thus, in view of the uniqueness part of the preparation theorem (see 1.4), h must be equal to q. Therefore, if a distinguished germ divides in On a germ from Qn, then the same is true in Qn. 2a. A distinguished germ of positive degree is reducible in On if and only if it is a product of distinguished germs of positive degree.

Indeed, suppose that a distinguished germ c is the product of germs g1 and g2 which are non-invertible in On. Then g1, g2 are regular and of positive order (see nO 1 and 1.4). By the preparation theorem, we have gi = hici, where hi are invertible in On. However the Ci are distinguished and of positive (14) The right hand side is defined as ranka(ih, ... ,[Jk), where the 9i'S are representatives of the gi'S.

e5) The only distinguished germ of degree 0 is 1.

146

1.2.2b

1. Rings of germs of holomorphic functions

degree. Thus c = (h 1h 2)(CIC2). Because of the uniqueness in the preparation theorem, c = CIC2.

2b. A distinguished germ is reducible in On precisely when it is reducible in Qn. To see this, take a distinguished germ C E On. We may assume that c is of positive degree (for otherwise c = 1 is irreducible in both On and Qn). Now, in view of 2a, if c is reducible in On, then it is also reducible in Qn. Conversely, if c is the product of two non-invertible germs in Qn, we may assume that they are monic, since the product of their leading coefficients is 1. Then they are distinguished of positive degree and hence are non-invertible in On (see nO 1). 3.

PROPOSITION.

The ring Oa i.3 a unique factorization domain.

PROOF. It is enough to consider the ring On. Now, the ring 0 0 = C is a unique factorization domain (see A. 6.1). Let n > 0, and assume that the ring On-l is a unique factorization domain. Then, by the Gauss theorem (see A. 6.2), the ring Qn - being isomorphic to On-l[X] - is a unique factorization domain.

In view of A. 9.5, it suffices to show that if an irreducible germ f E On divides the product of germs g, h E On, then it must divide one of them. By the preparation theorem (see n° 1), we may assume that the germ 1 is distinguished and g, h E Qn (see 1.4 (16)). But then the germ 1 is also irreducible in Qn (see n° 2a) and is a divisor in Qn of the product gh (see n° 1). Therefore f is a divisor in Qn of g or of h, and so the same is true in On. REMARK.

We have also proved that the ring Q is a unique factorization

domain. As a corollary we have the following property: In the ring On, every non-zero non-invertible germ 1 has a decomp0.3ition

f = 1;1 ... l;r, as.3ociated

e

where k i

7 ).

>

°

and the

Ii '.3

are irreducible and mutually non-

According to the proposition from A. 6.3, we get (see 1.5) the corollary: The discriminant of a monic germ p E Qn is zero if and only if p zs divisible by the square of a monic germ of positive degree.

This implies that: (16) For if x: On ~ On is an automorphism, then, if f is irreducible, so is xU), and if xU) divides the product X(g)x(h), then f divides the product gh. (17) It is sufficient to notice that if a germ g E On is invertible and k for some invertible germ h E On.

> 0,

then g

= hk

§3. The Preparation Theorem, Tom-Marinet version

147

A regular germ from On has no multiple factors precisely when the discriminant of its as/JOciated distinguished germ (see nO 1) is non-zero. (See A. 6.1, 1.4, and nO 1.)

In particular, The discriminant of a distinguished irreducible germ (see n° 2b) is nonzero.

4. Let pEOn be a distinguished germ. If the degree of P is positive, then P can be represented in a unique fashion as p = p~' ... p~r , where the Pi's are distinguished, irreducible (and hence of positive degree; see nO 1), mutually distinct, and the k;'s are all positive. Then the discriminant of the germ p is non-zero precisely when kl = ... = kr = 1. Indeed, p = ql ... q., where the germs qj are irreducible in Qn. We may assume that they are monic (as the product of their leading coefficients is 1). Then they have to be distinguished (see n° 1). This yields the above decomposition and proves its uniqueness (see nO 1). We define red p = Pl ... Pr, and, in addition, let red P = 1 if P = 1. For any regular element f E On, we define red f = red p, where P is the distinguished element associated with f (via the preparation theorem). Accordingly, the discriminant of red f is always non-zero (18).

§3_ The Preparation Theorem in Thom-Martinet version LEMMA . Let M be a finite module over On+k, and let N be finitely generated submodule of M regarded as a module over On. Then !vI = N + mnM implies that M = N. PROOF.

By the Nakayama lemma (see A. 10.2), it is enough to show the

implication: If A1 = N

+ mnM,

then M is finite over On .

Suppose k = 1. Let m = N + mnM. Then M is finite over the ring S = On + mnOn+l. In view of the Mather-Nakayama lemma (see A. 1.16), (18) Obviously, the zero sets of representatives of the germs f and red f coincide in a neighbourhood of zero. Then the zero set of a holomorphic function t= 0 is - in a suitably chosen local coordinate system - the zero set of a distinguished polynomial with non-zero discriminant.

148

1.3

I. Rings of germs of holomorphic functions

there exists a germ 7] = z~+ + lIZ:+-; + ... + Ir, where Ii E S, such that 7]M = O. Now, the germ 7] is regular (since the li(O, ... , 0, zn+d are constant germs). Denote its order by p. Let mI, ... , ms be generators of the module Mover On+I. If x E M, then x = L~ fimi, where fi E On+l, but the preparation theorem in division version gives fi = gi7] + L~':~ aij(zn+d j , where gi E On+l and aij E On. Hence x elements (Zn+l)jmi (i

=

Lij aij(zn+J)jmi. Thus the

= 1, ... , 8;j = O, ... ,p -1)

generate M over On.

Now let k > 0 and let us assume that our implication holds for k - l. Suppose M = N +mnM. Then M = On+k-IN +mn+k-IM. Since On+k-IN is a finitely generated submodule of M regarded as a module over On+k-l, the already verified implication (*) for k = 1 (with n + k - 1 replacing n) shows that the module M is finite over On+k-l. Hence, by the induction hypothesis, M is finite over On. COROLLARY. Assume the hypothesis of the lemma, and let L be a .'lUbmodule of M. Then M = L + N + mnM implies M = L + N. Indeed, it is sufficient to apply the lemma to the images under the natural homomorphism M ---t M / L of On-modules. Denote by 0 v the ring of germs at 0 of functions that are holomorphic with respect to the variables v = (Zn+l, ... , Zn+k). If 1 is an ideal of the ring OnH, then 1(0, v) = U(O, v) : f E I} is an ideal ofthe ring Ov' It is equal to the image of the ideal 1 under the epimorphism X : OnH :;) f ---t f(O, v) E Ov (see l.2). THE PREPARATION THEOREM IN THOM-MARTINET VERSION. Let 1 be an ideal of the ring On+k, and let aI, ... , a r E On+k. Then r

r

PROOF . By considering the images under X, one can see that the left hand side follows from the right hand side. Now note that, according to Hadamard's lemma (see C. 1.10), we have ker X = mnOn+k. Assuming the equality on the left hand side, we have x(I + L; Ona;) = Ov, and so (19) This version implies the division version (except for the uniqueness property). Indeed, if a germ

I E On is regular of order k, then (taking

v = zn+d we have Ov =

Ovl(O, v) (in view of Taylor's formula - since 1(0, v) generates "k-l

DO

.

On_IZ~

+ Onl·

mn.

L~:OI Cvi +

Therefore On

§3. The Preparation Theorem, Tom-Marinet version

r

On+k

=

1+

L

On a;

+ ffinOn+k

149

(20).

1

Thus the corollary implies the equality on the right hand side above.

eO) If X: M - - t N is an epimorphism of commutative groups, E C M, and X(E) then M = E+ ker x.

= N,

CHAPTER II

ANALYTIC SETS, ANALYTIC GERMS, AND THEIR IDEALS

§1. Dimension 1. Let M be a complex n-dimensional manifold. \Ve define the (complex) dimension of a subset E c ~M by the formula

dim E

= sup{ dim r:

r is a submanifold contained in E}

e).

In the case E is a submanifold, this definition is consistent with the one used previously (see C. 3.7). If E c N c M, where N is a submanifold, then dim E does not change when E is regarded as a subset of the manifold N. Biholomorphic mappings preserve the dimension: if h: !vI ---+ N is a biholomorphic mapping between complex manifolds AI, N, then dim h( E) = dim E for any E C AI. n -

We define also the codimension of a subset E C AI by codim E dimE.

=

2. Clearly, dim 0 = -OJ. The dimension of any countable non-empty set is equal to o. Note that dim E = n s, is a combination of those s forms. Thus 5 n U c 5 n U, where 5 = {IiI' = 0: i = 1, ... ,k, v = 0, ... , s} is an algebraic cone, and

5 c

5 (see B. 5.1). Now, let z E 5. There is an E > 0 such that if It I < E, then tz E U, and hence Ii(tz) = 2.:::0 f;v(z)tV = O. Consequently, Iiv(z) = 0, which means that z E S. Therefore 5 = 5 is an algebraic subset defined by homogeneous polynomials. so

4. A subset Z of a manifold M is called an analytic subset (of M or in M) if its germ at any point of the manifold Jo.!I is analytic or, equivalently, if every point of the manifold NI has an open neighbourhood U such that the set Z n U is a globally analytic subset of U e). In particular, any closed submanifold of the manifold ]1.1 is an analytic subset. Any subset of an analytic subset which is closed and open in the induced topology is analytic. The union and the intersection of a locally finite family of analytic subsets of (8) More generally, the implication is true if C is a simple germ; see IV.3, prop. 2. (9) Obviously, a globally analytic subset of !v! is analytic in itT. Generally speaking, the converse is not true. For instance, the only globally analytic subsets of a compact connected manifold Mare 0 and M. This is so because, owing to the maximum principle (see C. 3.9), every holomorphic function on !vI is constant.

156

II. Analytic sets

11.3.5

M is an analytic subset of M. The Cartesian product of analytic subsets of the manifolds M and N, respectively, is an analytic subset of the manifold M x N eO). If f: M --+ N is a holomorphic mapping of manifolds, then the inverse image of an analytic subset of N is an analytic subset of M. If N C M is a submanifold and Z is an analytic subset of M, then Z n N is an analytic subset of N. If N C A1 is a closed submanifold, then a subset of N is analytic in N if and only if it is analytic in M. If {C t } is an open cover of the manifold M, then a set Z C A1 is analytic in M precisely when, for each L, the set Z n C t is analytic in Ct. Every analytic subset is closed (see B. 1). Analytic subsets of open subsets of the manifold M are called locally analytic subsets of M (or in M). Therefore, a set Z c M is a locally analytic subset (of M) if and only if its germ at any of its points is analytic or, in other words, if each of its points has an open neighbourhood U such that Z n U is globally analytic in U. In particular, every submanifold of the manifold M is a locally analytic subset of A1. For subsets Z, M, we have the equivalence

(Z is analytic)

-¢=::>

(Z is locally analytic and closed) .

Any open subset in a locally analytic subset is locally analytic. If N C M is a submanifold, then: if Z is locally analytic in M, the set Z n N is locally analytic in N. If ZeN, then

(Z is locally analytic in N)

-¢=::>

(Z is locally analytic in M) .

The Cartesian product of locally analytic subsets of the manifolds M and N, respectively, is a locally analytic subset of the manifold M x N (11). If f: M --+ N is a holomorphic mapping of manifolds, then the inverse image of a locally analytic subset of N is a locally analytic subset of M. The intersection of a locally finite family of locally analytic subsets is a locally analytic subset (whereas the union of two or more locally analytic subsets is not necessarily a locally analytic subset). Clearly, every locally analytic subset is locally closed. If V and VV are non-empty subsets of the manifolds M and N, respectively, then:

(V x W is (locally) analytic in M x N) -¢=::>

-¢=::>

(V and HI are (locally) analytic in M and N, respectively)

(For, e.g., V is the inverse image of the set V x HI under the mapping M 3 z --+ (z, b) EM x N, where b E W.) eO) The same holds for any finite number of factors. (11) The same holds for any finite number of factors.

157

§3. Analytic sets and germs

If f: }vI - - - t N is a surjective submersion, the manifolds M and N are of dimension m and n, respectively, and ZeN, then

(Z {=::>

is a (locally) analytic subset (of constant dimension))

{=::>

(J-l(Z) is a (locally) analytic subset (of constant dimension)).

In this case dimrl(Z) = dimZ

+ (m -

n),

provided that Z =I 0. This follows directly from the definition of a submersion (similarly as in C. 4.2). Naturally, all biholomorphic mappings between manifolds preserve (local) analyticity of subvarieties.

5.

PROPOSITION.

Every nowhere dense analytic subset Z C M is thin in

M. This is a consequence of the following lemma: LEMMA. If Z is a representative of a non-full analytic germ at a point a E M, then there is a coordinate system 0, since the set H is connected (see B. 3.2 and 3.6). The function Pi is well-defined by the formula

where

It is a monic polynomial whose coefficients are locally bounded near {.6 = O} (see B. 5.3). They are holomorphic in H. Indeed, if Uo E H, then 7r;I(uO) = Uo X {w~, .. . ,w~;} and (by the implicit function theorem) there exist holomorphic functions WI, ... ,W r ; on an open connected neighbourhood U of the

point un, such that w,,(uo) = w~, P(u,w,,(u)) = 0 for u E U, and the w,,(u)'s are mutually distinct. Their graphs, being connected, must be contained in Ai, and hence for u E U we have 7r;l(U) = U X {Wl(U), ... ,wr;(u)}. In other words Pi(z) = (Zn - Wl(U)) ... (Zn - wr;(u)). Since the set {.6 = O} is thin in B (see 3.5), the coefficients extend to holomorphic functions on B, which means that Pi has an extension Pi that is a monic polynomial of degree ri with coefficients which are holomorphic in B. Obviously, P = P1 P2 in H x C, hence P = P I P2 in B x C, which implies that the germ p is the product of two monic germs of positive degrees (see 1. 1.5). Thus p is reducible (see 1. 2.1 ).

We are going to prove the following property If non-zero germs

J, g

E Oa are relatively przme, then dim(V(J) n

V(g)) < n - 1. In fact, we may assume tha.t the germs J, 9 are non-invertible and also (see 4.1 and A. 6.1) that they are irreducible. Furthermore, we ma.y assume that Oa = On and (in view of the preparation theorem) that the germ 9 is distinguished (see 1. 1.4 and 4.4). Let P be a representative of the germ 9 chosen as in the lemma. We may take the neighbourhood W = B x {Iznl < c} arbitrarily small and such that it contains the set {P = O} (see C. 2.2). In

168

11.5.3

II. Analytic sets

particular, we may assume that the germ I has a representative that is holomorphic in W. Suppose now that dim(V(f) n V(g)) = n - 1. Then there exists a non-empty submanifold r c {F = O} n {P = O} of dimension n - 1. Now, dim{6. = O} S n - 2 (see 1.2), and hence the set {P = 0, 6. = O} is of dimension S n - 2, as the fibres of its natural projection onto C n - 1 are finite (see 1.4). Therefore the above set cannot contain the submanifold r. Consequently, the set ra = rnA, open in r, is non-empty. It is an (n - I)-dimensional submanifold, and so it is open in A. As F vanishes on r a , it also vanishes on A and on {P = O}. Thus the germ I vanishes on V(g), and hence, by the theorem in nO 2, it is divisible by g, which is impossible. If iI, ...

, h EGa,

then

V(iI) n ... n V(fk)

= V(g) U B

where 9 is the greatest common divisor of the germs dimension S n - 2.

,

Ii

and B is a germ of

Indeed, omitting the trivial cases when one of the germs f; is invertible or iI = ... = h = 0 and removing the zero germs, we may assume that the germs f; are non-zero and non-invertible. Let gl, ... , gl be all distinct (up to association) irreducible divisors of the germs iI, ... , Ik. Then V(fd = U{V(gj): gj is a divisor of I;}. It follows that V(fd n ... n V(h) is the union of all the germs of the form V(gO\) n ... n V(ga.), where gao is a divisor of f; (i = 1, ... , k). Those among them for which lY1 = ... = lYk coincide with the germs V(gj), where gj is a common divisor of the germs iI, ... ,Ik' and hence their union is the germ V(g) (see A. 6.1). The remaining ones are of dimension S n - 2, according to the previously described property. For an analytic germ A (at a), we have the equivalences:

(A is simple of dimension n - 1)

(A = V(f), where is irreducible).

I

e

1

E Ga )

(21) This, combined with the lemma, implies that every simple germ of dimension (n -1) has an arbitrarily small representative in which there is an open dense subset that is a connected (n - I)-dimensional submanifold. This follows also from proposition 1 in IV. 3 that characterizes irreducible germs (in view of corollary 3 from proposition 2 in IV. 2.8).

169

§6. One-dimensional germs

(A is of constant dimension n - 1) ~ (A = V(J), where (moreover,

I

I

E Oa is non-zero

can be chosen to be

(22)).

without multiple factors)

Clearly, the right hand sides imply the left hand sides (see nO 2, corollary 2; and n° 1). Conversely, suppose A is of constant dimension n -1. According to (*) (see 4.1), we have A = V(g) U B, where dimB :::; n - 2. Each simple component of B must be contained in V(g), for otherwise (see 4.6) it would be a simple component of A. Thus A = V(g). Obviously, the germ 9 is nonzero (and one can replace 9 by a germ that does not have multiple factors). Assume, in addition, that A is irreducible of dimension n - 1. By taking the decomposition of 9 into irreducible factors we conclude (see nO 2, corollary 2) that A = V(J) for some irreducible germ I E Oa. The second of the above equivalences can be restated as follows.

(A is principal)

~

(A is of constant dimension n - lor n) .

From the representation (*) we get the following equivalence for germs ft, ...

···,Ik: (ft, ... , Ik are relatively prime)

~

(dim(VUd

n ... n V(Ik)) :::;

Indeed, both sides are equivalent to the condition V(g) = Note also that, for non-invertible germs

I, 9 E Oa

0 (see

n -

2)

nO 1).

we have:

(J,g are relatively prime) ~ dim(V(J) n V(g)) = n - 2 . This is a corollary of the inequality codim(V(JJ) n ... n V(Jk)) :::; k , I.e.,

dim(V(Jd for any non-invertible germs nO 1) (23).

Ii

n ... n VUd) : : :

n - k ,

E Oa (see III. 4.6, inequality (*) below, and

(22) Then, in view of the theorem from nO 2, the germ f is unique up to association. (23) This inequality implies also that the germ v(f)n V(g) must be of constant dimension n - 2 (see IV. 3.1, the corollary from proposition 4 below).

170

II.6.1

II. Analytic sets

§6. One-dimensional germs. The Puiseux theorem 1. Let H (z, w) be a polynomial in w E C which is monic of degree p and has holomorphic coefficients in a neighbourhood of zero in C (see C. 2.1). Then the germ Ho E Q2 can be identified with a polynomial from 0 1 [T] (via the natural isomorphism 01[T] --+ Q2; see I. 1.5).

THE PUISEUX THEOREM (FIRST VERSION). If Ho is irreducible in 0 1 [TJ, then there is a holomorphic function on the disc n = {Izl < 8}, such that p-l

H(zP,w) =

II (w -

h(e2triv/pz))

m

n xC.

v=o

Moreover, if H is a distinguished polynomial, then h(O)

= O.

The following lemma will be used in the proof of the theorem. LEMMA. If a bounded holomorphic function 1] defined on the half-plane P = {imz > ,8}, where ,8 E R, satisfies the condition 1](z + 1) = 1](z) in P, then 1](z) = ,(e 2rriz ) in P for some holomorphic function, on

{Iwl
0 such that the coefficients of the polynomial H are bounded and holomorphic in the disc {Izl < (2} and D(z) i- 0 in the annulus {O < Izl < d· Thus ~~(z,w) i- 0, provided that 0 < Izi < (2 and H(z,w) = 0 (see C. 2.1). Hence the function G(t,w) = H(e 2trit ,w) is holomorphic in P x C, where P = {imt >,8} and e- 21r ,B = (2. It is a monic polynomial of degree p the coefficients of which are holomorphic and bounded in P; we have ~~ (t, w) i- 0 in the set Z = {G = O} and all the roots ofthe polynomial w --+ G(t, w) are distinct for each t E P. So, in view of the implicit function theorem, the set Z is a locally topographic submanifold of P x C. Since the natural projection 7r: Z --+ P is proper (see B. 5.2 and 3), it is a finite covering (see B. 3.2 proposition 1) of multiplicity p.

=

(24) For if Wo = e 27r ;zo, Zo E P, then e 27r ;((w) w in U and «(U) C P, where ( is a holomorphic function in a neighbourhood U of the point Zo. Hence I( w) = ry( «( w)) in U.

171

§6. One-dimensional germs

Any topological component ( of the submanifold Z is also locally topographic and the restriction 7r(: (-----t P is also a covering (see B. 3.2). It must be one-sheeted (see B. 3.2, proposition 2), so ( is a holomorphic function in P. Therefore Z = TJl U ... U TJp, where the TJi are mutually distinct holomorphic functions on P. Consequently, G(t,w) = ITi(w - TJi(t)) in P x C and the functions TJi are bounded (see B. 5.3). Let A = {TJl, ... ,TJp}' Each of the functions TJi(t + I), I E Z, belongs to A, because Z - (l, 0) c Z, hence each equivalence class ofthe relation "TJi(t+l) = TJj(t) in P for some l E Z" must be of the form {TJ,,(t), TJ,,(t + 1), ... ,TJ,,( t + p" - 1 Moreover, TJ,,(t + p,,) = TJv(t) in P. Changing the order of indices, we get

n·

This means that G(t,w)

TJs(t

+ v))

= IT:=1 Gs(t,w),

=

where Gs(t,w)

IT~~";/(w­

in P x C. Now, the coefficients of the polynomial G s satisfy the hypotheses of the lemma (since Gs(t + 1,w) = Gs(t,w)). Hence Gs(t,w) = Hs( e2rrit , w) in P x C, where Hs is a polynomial of degree Ps > 0 with holomorphic coefficients in the disc {Izl < e}, and we have H = HI ... Hk in {Izl < d. It follows that k = 1 and PI = p, for otherwise Ho would be reducible in 0 1 [T] (see A. 1.13 and A. 2.3). Putting TJ = TJl, we obtain

G(t,w) = IT~:~(w - TJ(t + v)) in P x C. On the other hand, the function ~P = {imt > fJ/p} :1 t -----t TJ(pt) satisfies the hypotheses of the lemma, and therefore TJ(pt) = h( ehit ) in ~ P, where h is a holomorphic function on the disc {Izl
0, the arc {(t,h(t)) : 0:::; t :::; r} is simple and of class C1, because the limit limhO

1t(pt P - 1 ,h'(t))/r- 1

exists and is different from

zero. Here r = min(p, P2, ... ,]In) and the pj are the multiplicities of the zeros at 0 of the components of the mapping h.

e

1 ) It is enough to assume t.ha.t dim" V 2: 1, since then V" contains a one-dimensional analytic germ. (See e.g. the proof of the lemma in IV. 4.3.)

CHAPTER III

FUNDAMENTAL LEMMAS

§1. Lemmas on quasi-covers 1. Let X be a complex vector space, and let G be a finite subgroup of the group of linear au tomorphisms of X. A subset Z of the space X (respectively, a mapping f defined on X) is said to be invariant with respect to G or Ginvariant if 'P(Z) = Z (respectively, f 0 'P = f) for each 'P E G. LEMMA O. Every G-invariant algebraic subset of X can be defined by G-invariant polynomials.

Indeed, let G = {'PI, ... , 'P L}, and let Z c X be a G-invariant algebraic subset defined by the polynomials II, .. . ,ik. Then

Z = {gij

= 0;

i = 1, ... , k; j = 1, ... , l} = {Fiv = 0; i = 1, ... , k;

v = 1, ... , l} ,

where gij = fi 0 'P j, Fiv = 0' v 0 (gil, ... ,gi/), and 0'1, . . . ,0', are the basic symmetric polynomials of 1 variables (see B. 5.3). Hence it is enough to check that the polynomials Fiv are G-invariant. Let us take an arbitrary 'Ps. As. the mapping G :3 'P - - 4 'P 0 'Ps EGis bijective, we have 'Pi 0 'Ps = 'P{Jj' where j = 1, ... , k and (;31,' .. ,;3,) is a permutation of the set {I, ... , l}. Thus gij 0 'P s = gi{Jj and we get Fiv 0 'P s = 0' v 0 (gi{J, , ... ,gi{J/) = Fiv. In particular, if X and Yare vector spaces and pEN \ 0, then the mappmgs

§l.

179

Lemmas on quasi-covers

where a = (aI, ... ,a p ) are permutations of the set {I, ... , p}, form a finite subgroup of the group of linear automorphisms of the vector space XP x Y. Subsets of this space and mappings defined on this space which are invariant with respect to the above subgroup are said to be symmetric with respect to x = (Xl' ... ' Xp). In other words, a set Z C XP x Y (respectively, a mapping f defined on XP x Y) is symmetric with respect to X if for every permutation a, IIa(Z) = Z (respectively, f 0 IIa = J). Lemma 0 yields the following LEMMA 1. If a set Z C XP x Y is algebraic and symmetric with respect to x, then it can be defined by polynomials that are symmetric with respect to x, i. e., there exists a polynomial mapping P: XP x Y - - t C· that is symmetric with respect to x and such that Z = P-l(O).

Let X be a vector space. A polynomial mapping P( 'T/l , ... , 'T/p, v) on the space Xp+l with values in a vector space is called a collector if it is symmetric with respect to 'T/ = ('T/l, ... ,'T/p) and

p-l(O) = {v = 'T/d u ... u {v = 'T/p} For example, the polynomial Cp+l :3 ('T/, v) - - t (v - 'T/d ... (v - 'T/p) E C is a collector. It follows from lemma 1 that for every vector space X and pEN there exists a collector P: Xp+l - - t C'. LEMMA 2. There is a collector P: Xp+l ('T/l, ... , 'T/p) E XP satisfying the condition 'T/i d1)l

(v

--t

cr such that, for each 'T/ = 'T/I for i > 1, the differential

--t

of.

P( 'T/, v)) is injective.

PROOF. Take an arbitrary r 2: n = dim X. There exist linear forms 'PI, ... ,'Pr E X*, such that every set of n of these forms is linearly independent. (They can be chosen by induction: if a sequence 'PI, ... ,'Pk E X*, where

k 2: n, has this property, we take 'PHI from the complement of the union of all hyperplanes generated by n - 1 elements of this sequence (1).) Observe that the following property holds when aij E C for i E I and j E J, where I and J are finite sets:

(#I> (#J)(n - 1),

II

aij

for i E

I) =? (#{i:

aij

= O} 2: n

jEJ

for some j E (Otherwise, since I C Uj{i: Assume that r

e)

aij

= O},

J).

we would get #1 ~ (#J)(n - 1).)

> pen -1). Then P = (Pl, ... ,Pr ), where Pi('T/,v) = 'Pi(V-

This is possible, as hyperplanes are nowhere dense.

180

III. 1.2

III. Fundamental lemmas

TJ1)' .. 'Pi(V - TJp), IS a collector. Indeed, by (*), the equation P(TJ, v) = 0 implies that 'Pi( v - TJj) = 0 for some j and for n distinct values of the index i, and hence v = TJj. Now fix TJ such that TJi i- TJl for i > 1. We have to show that the differential d"" P,., is injective, where P,., = (P"", ... , P"'r) and P,.,i : v ----; 'Pi(V - TJJ) ... 'Pi(V - TJr). We have d""P"'i = Ci'Pi, where Ci = It>l 'Pi(TJl - TJj)· Observe that #{i: Ci = O} ::; (p - 1)(n - 1). For if this were not the case, then, in view of (*), there would exist an index j > 1 such that 'Pi(1]l - 1]j) = 0 for at least n values of the index i, and hence TJi would be equal to TJ1, contrary to our assumption about TJ. Thus, if we assume that r 2:: (p - 1)( n - 1) + n, then Ci i- 0 for at least n distinct values of the index i, which implies that the differential d,." P,., = (Cl 'PI, ... ,cr'Pr) is injective. 2. Let J..{ be a connected manifold, and let X be a vector space. By a quasi-cover in the Cartesian product M X X we mean a pair (Z, A) in which Z is a thin subset of M, and A is a closed locally topographic submanifold of (M \ Z) x X (see C. 3.17) such that the natural projection if: A ----; M is proper (or, equivalently, each point of the manifold M has a neighbourhood U such that the fibres Au = {v EX: (u,v) E A} (2), for u E U, are uniformly bounded (see B. 5.2)). If (Z, A) is a quasi-cover, then the natural projection 1T: A ----; M \ Z is a finite covering (see B. 3.2, proposition 1; and B. 2.4). Since M \ Z is connected (see II. 2.4), then the covering has a multiplicity (see B. 3.2), say k, which is called the multiplicity of the quasicover (Z, A) and we say that the quasi-cover is k-sheeted. The set A is called the adherence of the quasi-cover. Notice that if (Z,A) is a quasi-cover in A1 x X, then so is (Z,A') for any closed and open subset A' of A; in particular, it is true when A' is a component of A. Moreover, (Z n G, Ac) is a quasi-cover in G x X, where G is an open and connected subset of M. 3. Let (Z,A) be an s-sheeted quasi-cover in the product AI x X, where

s> O. 3. If R( TJl, ... ,TJp, v) is a polynomial mapping on Xp+1, with values in C , which is symmetric with respect to 1] = (TJ1, ... , TJp), then there is a unique holomorphic mapping H M x X ----; C q such that for any (u,V)E(M\Z)xX, LEMMA

q

H(u,v)=R(TJ1, ... ,TJp,v),

where

{TJ1, ... ,TJp}=A u

.

(2) Throughout the book we will be using the following notation. If Z C M x N, then = {y EN: (x, y) E Z} for x E M, and ZE = Z n (E x N) = 1r- 1 (E) for E C M,

Zx

where

1r:

Z

--+

N is the natural projection.

§l.

181

Lemmas on quasi-covers

Indeed, furnishing X with a linear coordinate system, we can write = E.a.(T/)v·, where a.: XP - - t C q are symmetric polynomial mappings. Then the mappings c~ : M\ Z - - t C q , well-defined by the formula

R(T/,v)

c~(u) = a.(T/l, ... ,T/p) for U E M \ Z, where {T/l,"" T/p} = Au, are holomorphic and locally bounded near Z. Therefore they have holomorphic extensions c.: M - - t c q (see II. 2.1). The mapping defined by H(u,v) = c.(u)v· has the required property. The uniqueness of H follows from the fact that (M \ Z) x X is dense in M xX.

E.

PROPOSITION 1. The projection 7r

function M :3 U - - t #Au ~ p e)·

A --t

M is open. Therefore the #Au is lower semi-continuous (see B. 2.1) and finite:

THE FIRST LEMMA ON QUASI-COVERS. The adherence A is analytic (4) in M X X. If P: Xp+l - - t C r is a collector, then there exists a unzque holomorphic mapping F = Fp: M X X - - t C q such that

F(u,v)

= P(T/1, ... ,T/p,v), where hI, ... ,T/p} = Au, for(u,v)E(M\Z)xX.

Then A = F-l(O). PROOF of proposition 1 and the first lemma on quasi-covers. The exist.ence and uniqueness of the mapping F follows from lemma 3. Let ii- : F-l(O) - - t M be the natural projection. Then ii--1(M \ Z) = A. It is sufficient to show that ii- is open because, from the fact that M\Z is dense, we have A = F- 1 (0) (see B. 2.1), and consequently, 7r = ii-. If it were not true, there would exist a compact neighbourhood U x V of some point (uo, vo) E F- l (0) and a sequence Uti

--t

Uo such that Uti

1. ii-(F-l(O) n (U

x V)); moreover,

as Z is nowhere dense, one can require that Uti 1. U \ Z. Then we would have p(T/r, ... ,T/~,v) = F(utl,v) f:. 0 for each v and v E V, where "Ii E Au •. This would imply 1Jr rf. V. In view of the uniform boundedness of the fibres Au", one could choose convergent subsequences "If" - - t "Ii f:. Vo. Taking the limit, we would get F( Uo, vo) = P( "11, ... , T/p, vo) f:. 0, in contradiction with the choice of (uo, vo). According to lemma 1, there is a polynomial mapping Q( "11, ... , T/p, v) of Xp+1 to c· which is symmetric with respect to T/1, ... ,T/p and such that

Q-l(O) = U{u = "Ii = T/j} . ioFj

e)

This is because M \ Z is dense and #Au ~ p.

(4) It is even globally analytic.

182

III.2.1

III. Fundamental lemmas

By lemma 3, there exist a holomorphic mapping G: M x X that

G(u, v) for

= Q(TJI,"" TJp, v),

where

~

cs

such

{r;1,"" TJp} = Au,

(u,v) E (M \ Z) x X.

We have the following lemmas

4. If at a point (uo, vo) the adherence ifold, then G(uo,vo) i- O. LEMMA

A is

a topographic subman-

5. If P : Xp+1 ~ C r is a collector, F = Fp, and the adherence A is a topographic submanifold at a point (uo, vo), then there are TJI, ... , TJp E X such that TJi i- TJI = Vo for i > 1, and LEMMA

F(uo,v)

= P(T]I, ... ,TJp,v)

for some

v EX.

PROOF of lemmas 4 and 5. We can choose neighbourhoods U and V of the points Uo and Vo, respectively, in such a way that the set An (U x V) is the graph of a continuous mapping 0,

and

§2. Regular and k-normal ideals and germs

187

or an analytic germ at 0 E C n is k-normal or k-regular, it is also k-normal or k-regular, respectively, both in the coordinate systems 'P x e" and e' x 1/;.) Indeed, it is enough to verify the claim for ideals (see II. 4.4). Now, let

X = 'P x e" or X = e' X 1/;. Then Ch 0 X-I = Ok. The condition (4) of knormality of an ideal I (see nO 2) is equivalent to the condition On L:i Okgi+I for some gi E On. Thus, if I is k-normal, then, by passing to the images under the automorphism f --+ f 0 X-I of the ring On, we conclude that the ideal 10 X-I is k-normal. Moreover, if On n I = 0, then Ok n (I 0 X-I) = o.

In particular, this implies that: If a k-normal ideal of the ring On is r-regular after a linear change of the coordinates ZI, ... , Zk (i. e., in the coordinate system 'P x e", as above), then r :::; k. (See n° 5.)

7. Let 0 :::; k :::; n, and let e" denote the identity mapping of the space Cn-

k.

PROPOSITION . If ideals h, ... , 1m of the ring On, or analytic germs AI, ... ,Am at 0 E C n , are k-normal, then all of them are regular after any change of the coordinates ZI, ... , Zk that belong to a dense subset of Lo(C k , C k ), i.e., in any coordinate system 'P x e" for any 'P from a dense sub:Jet of Lo(C k , C k ).

It suffices to consider the case of ideals. For 0 :::; r < consider the condition: PROOF

(C r )

(8).

71,

The ideal I is T-normal or s-regular for some s :::::: r .

c

Denote by e r the identity mapping of n - r . Let T > O. In view of the previous property (see nO 6) and condition (1) for (T-1)-and T-normality (see n° 2), if the ideals II, ... ,1m satisfy the condition (c r ), then they satisfy the condition (cr-d in the coordinate system 'P x e r for any 'P from a dense subset of Lo(C r , C r ). (Indeed, each of those ideals which is r-normal but not r-regular contains a non-zero germ from Or; see nO 5. These germs are regular in any coordinate system 'P from a dense subset of Lo(C r , C r ); see 1. 1.4.) Therefore, from the dense subset of Lo(C k , C k ), one can choose successively k k 'Pk-I, ... ,'PI E Lo(C , C ) arbitrarily close to the identity mapping such that the ideals h, ... ,1m in the coordinate system ('PI 0 ... 0 'P k) X e" satisfy condition (co). Owing to the fact that the composition is continuous (see for any

'Pk

B. 5.2), it follows that the automorphisms 'P E Lo(Ck,C k ) for which the (8) The set of such 'P is open. Namely, it is the complement of a nowhere-dense algebraic subset of L(C k , C k ). (See corollary 2 from the Cartan-Remmert theorem in V. 3.3.)

188

III. Fundamental lemmas

III.3.1

ideals II, ... , I m in the coordinate system r.p x e" satisfy condition (co) form a dense subset. Finally, it is enough to observe that any O-normal ideal is O-or ( -1 )-regular. The proposition implies the following corollaries. Every k-normal analytic ideal (of On) or germ (at 0 E en) becomes rregular, with some r ::; k, after a suitable change of the coordinates ZI, ... , Zk. (See n° 6 and II. 4.4.) If M is an n-dimensional manifold and a E M, then arbitrary ideals II, ... , Im of the ring Oa or arbitrary analytic germs AI, ... ,Am at a are simultaneously regular in a suitable coordinate system at a (see II. 4.4). In the case when M is a vector space and a = 0, such coordinate systems that, in addition, are linear, constitute a dense set in Lo( M, en) (see footnote (8)). If A is a k-dimensional analytic germ at 0 E en (0 ::; k ::; n or k then: (A is k-regular) ~ (A is k-normal).

= -00),

Indeed, assuming that k ~ 0, if A is k-normal, then, after a suitable change of the coordinates ZI, ... , Zk, it becomes r-regular, where r ::; k. Thus dim A ::; r (see nO 4), and so k = r. Hence A is k-regular (see n° 6).

§3. Ruckert's descriptive lemma Let 0 ::; k ::; n. 1. Let I be a k-regular prime ideal of the ring On. Then the ring On is an integral domain (see A. 1.11 and 2.1) and Ok is factorial (see 1. 2.3 and 2.5). Thus, for each ij, j = k + 1, ... , n, we have a minimal polynomial

pj E Ok[T], where Pj E Ok[T] (see A. 8.2, 2.2, and 2.5). 1. The germs Pj(Zj), j = k + 1, ... , n, belong to the ideal I, are distinguished (in Odzj], respectively), and have non-zero discriminants. LEMMA

-

Indeed, since Pj(Zj) = pj(ij) = 0 (see 2.1), the germ Pj(Zj) belongs to I. It has a non-zero discriminant, because Pj is irreducible (see A. 8.2, the proposition in A. 6.3, and 1. 1.5). It is distinguished. Indeed, the preparation theorem (see I. 2.1) implies that it is associated with a distinguished germ qj(Zj), where qj E Ok[Tj. Thus qj(Zj) E I, which means that qj(i j ) = 0, and hence pj (as the minimal polynomial for i j) is a divisor of qj. Accordingly,

189

§3. Ruckert's descriptive lemma

the germ Pj(Zj) is a divisor in OdZj] of the distinguished germ qj(Zj), and so it is itself distinguished (see I. 2.1). LEMMA 2. The Weierstrass set W(PHl, ... ,Pn), where pj are representatives of the germs Pj(Zj) and have non-zero discriminants, can be made arbitrarily small. (See 1. 1.5 and 2.3.) RUCKERT'S DESCRIPTIVE LEMMA. Let A = V(l) be the locus of a kregular prime ideal I of the ring On. In particular, A can be an irreducible k-regular analytic germ at 0 E en (9). Then there exist:

an open connected neighbourhood

n

of zero in

an analytic nowhere dense subset Z in

ek,

n,

a representative V of the germ A, which is analytic in

n x en-

k

,

such that (1) the natural projection

(2)

7r-

1 (0)

7r:

V

~

n

is proper,

= 0,

(3) the set V' = VO\Z is a non-empty locally topographic submanifold of

n x en- k •

Thus (see B. 3.2, proposition 1; B. 2.4; and II. 3.6) the projection 7rv' is a finite covering of multiplicity> o.

V' ~

n\Z

Note that, in such a case, (Z, V') is a quasi-cover in

nx

Cn -

k

(see B.

2.4). Under the assumptions of the descriptive lemma, we have the following corollaries (notation as in the lemma). COROLLARY 1. The sets Vo', where D,' c n are open neighbourhoods of the origin in e k , form a base of neighbourhoods of zero in V, and 7r(Vo' ) = n'. Therefore the image under the projection onto k of any representative of the germ A is a neighbourhood of zero.

e

Indeed, in view of (1) and (2), the sets Vo' can be made arbitrarily small (see B. 2.4). The projection 7rvn , ~ D,' is proper and hence closed. Therefore, since 7r(Vf!'\z) 2.3).

= n' \

Z, we have 7r(Vo') :J

n'

(see B. 2.4 and B.

COROLLARY 1a. For any open neighbourhood D,' c n of zero, the triple Z n n', VO' satisfies the conditions (1) - (3). For any sufficiently small n', the set VO'\Z is dense in Vo'.

D,',

(9) See II. 4.6, proposition 2; and II. 4.5. From Hilbert's Nullstellensatz (see 4.1) it follows that these two conditions are equivalent.

190

III. Fundamental lemmas

IIL3.1

To see this, note that the first lemma on quasi-covers (see 1.3) implies that the set V = Vrl\Z

n (n

X

Cn-

k

)

is analytic in

n x Cn-

k

.

Also, the set

Vz is analytic in n x C - and we have V = V U VZi so, A = Vo U (Vz)o. But A i- (Vz)o by corollary 1. Hence A = Vo eO). Thus, in view of corollary 1, we have Vrl' C Vrl ' \Z, provided that n' is sufficiently small. n

k

COROLLARY 2. The multiplicity of the covering ?Tv' depends exclusively on the germ A and is independent of any linear change of the coordinates ZHI, ... ,Zn (11) (cf. 2.6).

C

k

In order to see this, take X = e' x 'l/J, where e' is the identity mapping of and'l/J E Lo(C n - k , C n - k ). Let A = X(A). Let V, V be representatives of

n

the germs A, A, respectively, and n, be neighbourhoods of zero, chosen as in the descriptive lemma. Note that xCV) is also a representative of the germ

A. By corollary 1, the sets Vrl' form a base of neighbourhoods of zero in V, whereas the sets X(V)rl' = X(Vrl ') form a base of neighbourhoods of zero in xCV). Therefore we must have Vrl' = X(V)rl' for some n' c n n Hence

n.

Vu = X(V)u = ¢(Vu) for u En'. Thus the multiplicities of the coverings ?Tv' and ?T t:n - being equal to # Vu and # Vu , respectively, for u E n' from the complement of a nowhere dense subset of n' - must coincide. PROOF

of the descriptive lemma. There is a system of generators

for the ideal I, where J; E Ch[XHl, .. . , Xnl (see 2.2). In view of lemma 1, we may assume that this system contains the germs Pj(Zj), j = k + 1, ... ,n. By lemma 2, there exist arbitrarily small open connected neighbourhoods of zero in neCk and fl. C C n - k , and a Weierstrass set W = W(PHl, ... ,Pn) C nx fl., where the Pj, which are representatives of the Pj(Zj), have coefficients that are holomorphic in n. We may also assume that the germs J;(Zk+I, .. . , zn) have representatives j; which are polynomials in v = (ZHb"" zn) with coefficients holomorphic in n. Then each of the germs Pi is equal to some

ii.

ir

Hence the set V = {il = ... = = O} is an analytic subset of that represents A and is contained in W. As a result,

(#)

n x C n- k

vcnxfl.,

eO) Due to the fact the germ A is irreducible - see footnote (9). In this chapter, corollary la will be used only in the proof of the classic descriptive lemma, which is not used in the proof of Hilbert's Nullstellensatz.

(11) It is also independent of any linear change of the coordinates

Zl, ... , Z k.

191

§3. Ruckert's descriptive lemma

the natural projection 1[: V ~ n is proper, and 1[-1(0)

=

0 (see 2.3 and

B. 2.4; we have 1j(0) = 0, for otherwise I = On). Now the set Wn\Z*, where

Z* = Z(Pk+l, ... ,Pn), is a locally topographic submanifold of n X C n- k (see lemma 2, and 2.3). Consequently, it is enough to show that for suitably chosen nand t:. there is an analytic set Z :J Z*, that is nowhere dense in nand such that

VI = Vn\Z =I- 0, { for each Z E VI, the germ V; contains a smooth germ of dimension k. For if z E VI, then we must have V; = Wz (otherwise, since VI C Wn\z* , one would have dim V; < k; see II. 3.3), and so Viis a topographic submanifold at z. According to the primitive element theorem for integral domains (see A. 8.3), there is a primitive element

wof the extension On of the ring Ok, where

wE On (see 2.2 condition (3)). Therefore 5ij = Qj(w), where DE Ok \0 and Qj E Ok[T] (j = k + 1, ... ,n). Thus DZj - Qj(w) E I, and so

where. aij E On. Since the element

wis integral over Ok (see (4) in 2.2), it has

a minimal polynomial G E Ok[T], where G E OdT]. Then the polynomial G is irreducible (see A. 8.2 and 2.5), and hence its discriminant Do E Ok is non-zero (see the proposition in A. 6.3). Thus we have G(w) G( w) E I. Therefore

=

0, and so

un where bj E On. For some m, we have

(see A. 2.2 and 2.1), and hence

G must

be a divisor of Fi (Qk+l,"" Qn)

in OdT] (see A. 8.2); i.e., Fi(Qk+l, ... ,Qn) =

CHi,

for some H j E Ok[T].

192

III. Fundamental lemmas

III.3.2

Therefore Fi(Qk+1, ... ,Qn) = GHi (see 2.1 and 2.5) and, substituting the germ t E Out ~ Ou = Ok of the function (u, t) ~ t, we obtain (see A. 2.2):

Fi (Qk+1(t), ... , Qn(t))

(6)

Now the neighbourhood W,

= G(t)Hi(t),

n X .0. can be

i

= 1, ... , r

.

chosen so small that the germs

aij, bi have representatives w, aij, h; which are holomorphic in

nx

.0., the

germs 6,60 have representatives 8 ¢ 0, 80 ¢ 0 which are holomorphic in n, and the coefficients of the polynomials Q j , G, Hi, F; have holomorphic representatives in n. Let Qj( u, t), G( u, t), H;( u, t), F;( u, v) be polynomials with respect to t and v, respectively, whose coefficients are holomorphic representatives in n of the coefficients of the polynomials Qj,G,H;,F;, respectively. Then the equalities (a )-( 6) imply

= L a;j(z)j;(z) r

8(u)zj - Qj(u, w(z))

(a)

n x.0.,

m

j

= k + 1, ... , n,

;=1 r

= F;(u,8(U)Zk+1'''. ,8(U)Zn)

n X C n- k , i = 1, ... , r,

(c)

5(u)m f;(z)

(d)

Fi (u,Qk+1(U,t), ... ,Qn(U,t))=G(u,t)Hi(U,t) m nxc,

m

i=l, ... ,r. Indeed, it is easy to check that the germs at 0 of the left and right hand sides of the equalities (a)-(d) are equal, respectively, to the left and right hand sides of the equalities (a )-( 6) (12). Note that

50

is the discriminant of

(12) For instance, one can take the polynomials Qj, G·, Hi E Oo[T], and Ft E OO[Xk+l, whose coefficients coincide with those of Qj, Gj, Hi, and Pi, respectively. Then after the identifications 0 0 C OOXIl. and 00 C Ooxc - the equalities (a)-(d), with (c) restricted to [2 X Cl., will become

... , Xnl

r

(a)

bZj - Qj(w) =

Laid;,

j = k

+ 1, ... ,n,

i=l r

(c)

bmfi =F/(bzk+l, ... ,bzn),

(d)

Ft(Qk+l(t), ... ,Q~(t»)=G·(t)Hi(t),

where

(z, t)

Zj

--+

i=l, ... ,r, i=l, ... ,r,

denote the functions [2 x Cl. :1 Z --+ Zj E C and t denotes the function [2 x C :1 t E C. Now, the images of the left and right hand sides of the above equalities

193

§3. Ruckert's descriptive lemma

G

(13).

Now we set Z = Z* U {bbo = O}. Then VI = {( u, v) E en:

E

U

n \ z,

}i( u, v)

= 0, i = 1, ... , r}

.

The set

A={(u,v,t)Ee n+1 uEn\Z, G(u,t) =0, b(u)zj=Qj(u,t), j=k+1, ... ,n} is non-empty (because G and also G are of degree > 0). In view of the implicit function theorem (see C. 2.1), it is a locally topographic submanifold of n x en-HI. Let 7r*: e n+1 :1 (z,t) ----7 Z E en. The relations (a)-(d) imply that VI = 7r*(A). (The inclusion VI =:J 7r*(A) can be derived from the equalities (c) and (d), while the inclusion VI C 7r*(A) follows from the relations (#), (a), and (b) with t = w(z).) Consequently, the set VI is nonempty and each of its points belongs to a k-dimensional submanifold which, in turn, is contained in this set (see C. 3.17). This completes the proof of the descriptive lemma. It follows from the above argument that by taking Z = {b = O} we get also VI = 7r(A). REMARK.

2. Using the second part of the primitive element theorem for integral domains, we can obtain a more precise description of the submanifold VI. First note that:

Under the hypothesis of the descriptive lemma (assuming k < n) and after a suitable change ofthe coordinates Zk+1, ... ,Zn (see 2.6), 2k+1 becomes a primitive element of the extension Indeed, the set x(C \ 0) C

Ok:

On

of the ring

Ok.

is infinite and hence there is a primitive

element of the form L:~+1 cji j , where Let us take the coordinate system 'P

Cj

E

e \ 0 (see A.

8.3 and (3) in 2.2).

= (ZI, ... ,Zk, L:~+l CjZj,Zk+2, ... ,zn).

Consider the ideal l' = 10'1'-1. We have the corresponding ring O~ = Onl1'. Its sub ring O~ and its elements

ij are the images of the ring Ok and the germs

under the homomorphisms Onxi). E f ---;. fo E On and Onxc " h ---;. ho E Out are, respectively, the left and right hand sides of the equalities (a)-(6). This is so because the images of the polynomials Q;, G* , H;' ,F;* via the induced homomorphisms are the polynomials Qj, G, Hi, Fi (see A. 2.2).

(13) Because 80 is the discriminant of G* (see A. 4.3 and 1. 1.5).

194

III. Fundamental lemmas

II1.3.2

Zj, via the natural epimorphism On ~ O~. Now, the ideal I' is the image

of the ideal I under the isomorphism On :3 f ~ f 0 cp-I E On, whereas the germ Z~+I is the image of the germ L~-I CjZj. Therefore we have the induced isomorphism

On

~ O~ under which the image of the subring Ok is

the subring O~, and the image of the element L~+I ejzj is the element Z~+I. This implies that Z~+I is a primitive element of the extension O~ of the ring

O~. Suppose now that Zk+I is a primitive element of the extension

On of the

ring Ok (and that the assumptions of the descriptive lemma are satisfied). Then G = Pk+I in the proof of the descriptive lemma. According to the second part of the primitive element theorem for integral domains (see A. 8.3;

On is finite over Ok, by (4)

in 2.2), we can set 5 = 50. Moreover, we may

assume that Qk+I = 5T. Now, taking Z = {8 = a}, we have V' = 7r*(A), by the remark that follows the proof of the descriptive lemma. But then

A

= {(u,v,t) E n x

en-k+I : 8(u)

-# O,Pk+I(U,t) = O,Zk+I = t, 8(u)zj

=

Qj(u,t),j

=

k+ 2, ... ,n},

and so V'

=

((u,v) E

nx

e n - k : 8(u)

-# O,Pk+I(U,Zk+I) = 0,8(u)zj = = Qj(u,Zk+J),j =

k+2, ... ,n}.

The implicit function theorem implies that the set V'is a non-empty locally topographic submanifold in n x e n - k . Therefore (taking into account corollary 1a) we have RUCKERT'S CLASSIC DESCRIPTIVE LEMMA.

of the descriptive lemma, with k

Suppo8e that the a88umption8

< n, are satisfied. Assume that Zk+I is a

primitive element of the extension On of the ring Ok; this can be achieved by a suitable change of the coordinates Zk+I, ... , Zn. Then there is a triple n, Z, V that satisfies the conclusions of the descriptive lemma, with V' dense in V and such that the following Ruckert formula holds: k (R) V' = Vn\z = ((u,v) E n x e n - : 8(u) -# 0, Pk+l(U,Zk+l) = 0, 8(u)zj

= Qj(u,zk+d,

j

= k+2, ... ,n}.

Here, Pk+I,Qk+2, ... ,Qn are polynomials in Zk+l whose coefficients are holomorphic in n. Moreover, Pk+l is a representative of the germ Pk+l(Zk+d

e

3a

)

e

3a ),

and is a distinguished polynomial with discriminant

The polynomial Pk+l is the minimal polynomial for Zk+l over

C\j

see nO 1.

8¢

0.

195

§3. Ruckert's descriptive lemma

Finally, Z

= {u

En: 8(u)

= O}.

e

4

)

REMARK 1. The above argument concerning the choice of a coordinate system n

'P =

(ZI,""

Zk,

L

CjZj, Zk+2,···, Zn)

k+1

for a k-regular prime ideal I shows (see the primitive element theorem in A. 8.3) that if ZHI,"" Zn C C \ 0 are infinite subsets, then there exist C j E Z j, j = k + 1, ... ,n, such that the ideal 10 'P -I satisfies the assumptions of the classic descriptive lemma. REMARK 2. Replacing t.he polynomials

Qi

by the remainders from their

division by PHI (see A. 2.4), one may assume that the degrees of the Qj are smaller than the degree of PHI, i.e., the multiplicity of the covering 7rv, (see the corollary below). COROLLARY. The multiplicity of the covenng 7rv' zn the descriptive lemma is equal to the dimension of the field of fractions of the ring Ok. In the classic descriptive lemma, it coincides with the degree of the polynomial PHI·

In fact, this is the case in the classic descriptive lemma. For if 8( u) =I- 0, then all roots of the polynomial t ----. Pk+ I ( 11, t) are distinct, and hence their number # V~ coincides with the dcgree of the polynomial PHI; the latter is equal to the dimension of the field of fractions of the ring

On over the field of

fractions of the ring Ok- This is so because PHI is the minimal polynomial (14) As a result, we have also (after reducing the size of 0) (#)

Vi

= {(u,v) E 0

X

Cn

-

k : .s(u)

=f. O,Pk+l(u,zk+Il =

0,

8Pk+l --(U,Zk+l)Zj = Rj(U,Zk+l), j=k+2, ... 8Z k +1

,n},

where Rj are polynomials in Zk+1 with coefficients that are holomorphic in 0. We may also assume (as in remark 2 below) that the degrees of Rj are smaller than the degree of Pk+1, i.e., smaller than the multiplicity of the covering 7l'V" Indeed, we have Pk+1 (ik+l )Zj = Rj (Zk+ J), where Rj E Ok[T), j = 2, ... , n. (See A. 8.3 footnote s ).) Thus Pk+l (Zk+l )Zj - (Zk+l) E I, and hence (after a suitable reduction

e

of size of 0)

~~::::: (u, zk+dzk

- Rj (u, Zk+1) = 0 on V, where Rj are polynomials with

respect to zk+1 with coefficients holomorphic in O. Here the Rj represent the coefficients of the polynomials Rj, respectively. Let V" denote the right hand side of the equality (#). Then Vi C V". But (except for the trivial case k = n - 1) the sets VI, V" are graphs of mappings of the set {(U,Zk+l EO xC: C n - k - 1 . Hence Vi V".

=

.s(u)

=f.

0,Pk+l(U,Zk+1) = O} to the space

196

III.4.1

III. Fundamental lemmas

for Zk+1 (see A. 5.3 and A. 8.2). Now the dimension is unaffected by any linear change of the coordinates Zk+l, ... ,Zn' Indeed, consider the ideal I' = 10 X, where X = e ' X 'P, e' is the identity mapping of e k , and 'P E Lo(en-k,en-k). Consider the corresponding ring

O~ = On/I' and its subring O~

- the image of Ok under the natural epimor-

phism On ---+ O~. The isomorphism On :1

f

---+

f

0

X-I E On, which maps

I onto I', induces an isomorphism On ---+ O~, mapping Ok onto O~. The mapping is such that its extension to the fields of fractions maps the field of fractions of the ring 1.15).

Ok

onto the field of fractions of the ring O~ (see A.

Thus (except for the trivial situation when k = n) the general case follows in view of corollary 2 of the descriptive lemma.

§4. Hilbert's Nullstellensatz and other consequences (concerning dimension, regularity, and k-normality)

L Let M be a complex manifold, and let a E M. HILBERT'S NULLSTELLENSATZ. For any ideal I of the ring Oa we have I(V(I») = rad I. If a germ f vanishes on V(I), then fm E I for same

m > 0 (15). In particular I(V(I)) = I if the ideal I is prime. (See A. 1.11). REMARK. This is obviously equivalent to the following classical version of Hilbert's Nullstellensatz: If f, g1, ... , gr are holomorphic function:! in a neighbourhood U of the point a and f = 0 on the set {gl = ... = gr = A}, then there is an exponent m > 0 such that fm = Cigi in a neighbourhood W C U of the point a for some holomorphic functions Ci in W.

L:;

PROOF of Hilbert's theorem. First suppose that the ideal I is prime. One may assume that M = en, a = 0, and I is k-regular for some k 2: 0 (see 2.7, I. 1.1 and II. 4.4). In view of corollary 1 from the descriptive lemma (see 3.1), we have

(15) The inclusion I(V(l) :::) rad lis trivial (see II. 4.3 and II. 4.5).

(16) For every germ from Ok n I(V(l)) has a representative which is independent of Zk+l, ... , Zn, vanishes on a representative of the germ V(l), and hence vanishes in a neighbourhood of zero.

197

§4. Hilbert's Nullstellensatz

Since I C I(V(I)) (see II. 4.5), it is enough to show the opposite inclusion. Now, let f E On \ I. Then j E On \ 0, and so jg E Ok \ 0 for some g E On (see A. 8.1, lemma 3.1; and (4) in 2.2). Thus fg E h + I c h + I(V(I)) for some h E Ok \ I. One must have h fg

f/. I(V(l))

in view of (*). Therefore

f/. I(V(I)) , and so f f/. I(V(l)).

Consider now an arbitrary ideal I. One may assume that I =I- Oa. Then I = II n ... n I k , where Ii are primary ideals (see A. 9.3). Since the ideals rad Ii are prime (see A. 9.3), we have I(V(I)) = ni I(V(I;)) = nJ(V(rad Ii)) pletes the proof.

= n rad Ii = rad I

(see II. 4.2-3 and A. 1.5), which com-

Consequently, we have the mutually inverse bijections A --+ I(A) and I ~ V(I) between the set of simple analytic germs at a and the set of prime ideals of the ring Oa (see II. 4.6, proposition 2; and II. 4.5). If the ideal I is primary, then the germ V(I) is simple (see II. 4.3 and A.9.3). If I is a proper ideal and II, ... ,Ir are all it.s i:wlated ideal.s, then V(l) = V(h)U ... UV(Ir) i.s the decompo.sition of the germ V(l) into .simple germs. (The equality follows from the irreducible primary decomposition of I; see A. 9.3 and II. 4.3. Next, we have V(Ii) ct V(Ij) if i =I- j, for otherwise the Nullstellensatz would imply Ii =:> Ij; see II. 4.2). If I =

h n ... n J r

is the irredundant primary decomposition, then V(I) = V(J 1 U

... U Y(J)r) is not necessarily the decomposition into simple germs. A counter-example is

furnished by the primary ideals J 1 = 02W, {w = O}o and Y(h) = o.

h = m 2 in the ring

02

for which V(h)

==

An ideal I of the ring 0" is k-normal, k-regular, or regular if and only if the germ V(l) is, respectively, k-normal, k-regular, regular. (This follows from the Nullstellensatz; see 2.4 and 2.5.)

2. Let 0 S k S n or k PROPOSITION.

=

-CXJ.

Any k-regular analytic germ at 0 E

en

is k-dimensional.

PROOF . One may assume that k ~ O. For an irreducible germ, the proposition follows, since dimA S k (see 2.4), from the descriptive lemma and corollary 1; for in this case each representative of the germ A must contain a k-dimensional submanifold. In the general case, the germ is of the form V(I), where I is a k-regular ideal of the ring On (see II. 4.5). Because k ~ 0, the ideal I is proper, and hence I = h n ... n J r , where Ji are primary ideals (see A. 9.3). Each of these ideals is k-normal (see 2.2) and so, after a suitable change of the coordinates Zl, ... , Zk, each of

198

III.4.3

III. Fundamental lemmas

them becomes ki-regular, where ki :S k (see the proposition in 2.7, and 2.6). Since the V (J i ) (after this change of coordinates) are simple and ki-regular, respectively (see nO 1), we have dim V(Ji) = k i . Therefore dim V(I) = max ki' because V(I) = V(JI ) U ... U V(J r ) (see II. 1.6 and II. 4.3). Now, if we had k i < k, i = 1, ... , r, then non-zero germs gi E Ch n Ji would exist (see (1) in 2.2); then 0 =I gl, ... , gk E I n Ok, contrary to the fact that I is k-regular. Hence k = max k i = dim Vel). Let M be an n-dimensional manifold, and let a E M. The proposition implies that: An analytic germ at a is k-dimensional precisely when it is k-regular in some coordinate system at a. If M is a vector space and a = 0, such coordinate systems (which, in addition, are linear) constitute a dense subset of Lo(M, en). (See 2.7.) An analytic germ at a is of dimension::; k if and only if it is k-normal in some coordinate system at a. This coordinate system can be linear if M is a vector space and a = o. (See 2.4 and 2.5.)

This implies the following HARTOGS' THEOREM. If Z c M is an analytic subset of dimension :S n - 2, then every holomorphic function on lvf \ Z extends to a holomorphic function on M. Indeed, for any a E Z, the germ Z a is (n - 2 )-normal in some coordinate system at a, and thus (see 2.4) the function f extends holomorphically across the point a (see II. 3.8 footnote (12)). REMARK. Clearly, Hartogs' theorem holds for holomorphic mappings on the set M \ Z with values in a vector space. The vector space cannot be replaced by a manifold. A counter-example is provided C2

by (7r \O)-1 : C 2 \ 0 (see VII. 5.2 below).

----> 112'

where 7r:

112 ---->

C 2 is the blow-up of C 2 at the origin

Note also that: If A is a k-regular analytic germ at 0 E en, then the image under the projection onto c k of any representative of the germ A is a neighbourhood of zero.

Indeed, by the proposition, the germ A is k-dimensional and therefore so is one of its simple components, say Ao (see II. 1.6). But Ao is also k-normal (see 2.4) and hence k-regular (see 2.7). Consequently, by corollary 1 from the descriptive lemma (see 3.1), our claim is true for A o, and hence also for A. 3. If M is a manifold and a E M, then, for any ideal I of the ring Oa,

199

§4. Hilbert's Nullstellensatz

we have the equivalences: codim 1 < 00 {:=}

{:=}

(1

:=)

dim V(I) ::; 0

rn 8 for some 8) {:=}

{:=}

V(I) Ca.

In fact, numbering these conditions, we already have the equivalence 2 (see I. 1.7), whereas the implications 2 ===;. 4 ===;. 3 are trivial. It remains to show that 3 ===;. 2. Now, if dim V(I) ::; 0, then, in some coordinate system at a, the ideal 1 is O-normal (see n° 2 and n° 1). Therefore the ideal 1 - in this coordinate system - contains (see (2) in 2.2) distinguished elements from e[ZiJ, (i = 1, ... , n = dimM). These elements will be of the form zji. 1

{:=}

Therefore the ideal will also contain the elements zP for Thus it contains the ideal rn 8 (see I. 1.7) (17).

Ipi = 8 = 81 + .. .+8 n .

In particular, for an analytic germ A at a we have: dim A ::; 0 (see II. 4.5). Hence

{:=}

A C a

A n analytic subset of a manifold !vI is of dimension::; 0 precisely when it is discrete. (See II. 1.5.)

The next characterization of systems of parameters for Oa also follows:

(h,.·., fn is a system of parameters of Oa) for

h, ... ,in

{:=}

(V(h, ... , fn) = a)

E Oa, where n = dimM.

Indeed, since dim Oa = n (by proposition 2 in I. 1.8), each side is equivalent to the condition: rn~ C I: Oafi C rna for some 8 (see A. 10.5). 4. Let N = {z E en: Zl = ... = Zk = O}, where 0::; k ::; n, and let A be an analytic germ at 0 in en. We have the following geometric characterization of k-normality: PROPOSITION.

(The germ A is k-normal)

{:=}

AnN C o.

PROOF. Set 1 = I(A). Let Ov denote the ring of germs (at 0) of holomorphic functions of the variables v = (Zk+1, ..• , zn); then 1(0, v) is an ideal of the ring Ov (see I. §3). Note that AnN = 0 x V(1(0, v)) (18). Now

(17) Since the implication 2 ==> 1 is trivial, instead of using the equivalence 1 ~ 2, it suffices to show 1 ==> 3. If dim V(l) > 0, then I' = 10'1"-1 is k-regular, where k > 0, for some coordinate system 'I" at a. Hence Ok n I' 0, and so codim I codim l' 00. Note also that the implication 4 ==> 2 follows from the Nullstellensatz.

=

=

=

(18) For take representatives F I , ... , Fr of generators of the ideal I. Then Fi(O, v) are representatives of generators of the ideal 1(0, v) and we have {FI Fr O} n N = Ox {F1 (0, v) Fr(O, v) OJ. Since A V(l) (see II. 4.5), it is enough to take the germs at 0 in the former equality.

= ... =

=

=

= ... = =

200

IlI.4.S

III. Fundamental lemmas

k-normality of the germ A, i.e., k-normality of the ideal I, means (see (4) in 2.2) that On = L: Okai + I for some ai E On. In view of the Thom-Martinet version of the preparation theorem (see I. §3), this is equivalent to the condition Ov = L: Ca;(O, v) + 1(0, v) for some a; E On, and hence to the condition codim I(O,v) < 00. This, in turn, is equivalent (see nO 3) to the inclusion V(I(O,v)) CO, hence to the inclusion An NcO. COROLLARY.

If dim A

= k,

then: (A is k-regular)

-¢=:::;>

AnN

= O.

(See

2.7.) The proposition implies also that: If AnN = 0, then a representative of the germ A is defined by polynomials with respect to Zk+l, • .. ,Zn whose coefficients are holomorphic in a neighbourhood of zero in C k . (For A = V(I(A)), and I(A) is k-normal; see

2.2.) 5. Let M be an n-dimensional vector space, and let A be a non-empty germ at O. The proposition in n° 4 implies the following formula

(#) codim A = max{dimN: N is a subspace such that NnA = O}

C

9

).

Indeed, if 0 ::; I ::; n, then the condition codim A ::::: I, i.e., dim A ::; k = n -I, is equivalent (see nO 2 and the proposition in nO 4) to the existence of a

linear coordinate system in which AnN = 0, where N = {Zl = ... = Zk = OJ. In other words, it is equivalent to the existence of an [-dimensional subspace N such that N n A = O. Notice that codim A NnA = O.

= dim N for any maximal subspace

N such that

Namely, in such a case, N = {Zl = ... = Zk = O} in some (linear) coordinate system with k = codim N. According to the proposition in n° 4, the germ A is k-normal, and therefore, after a suitable change of the coordinates Zl, ... ,Zk, it is r-regular, where 0::; r::; k (see 2.7; we have

e

9) It follows from formula (*) in nO 6 that for an analytic germ A ::f: 0 at a point a of the manifold M one has: codim A ~ I ¢::::? An C = a for some I-dimensional germ C which is analytic (or even smooth) at a (where O:S I:S dimM). This yields the formulae codim A

= max{ dim C: = max{dimC:

= a} = = a}.

C is an analytic germ at a such that C n A C is a smooth germ at a such that C n A

201

§4. Hilbert's Nullstellensatz

A i- 0). Thus dimA = r (see the proposition in nO 2) and, in view of the corollary from n° 4, we have No n A = 0, where No = {Zl = .,. = Zr = a}. Hence r = k, since N is maximal, which gives dim A = k.

If M

=

dim A

H

+N

is a direct sum of subspaces and AnN

= dim H

-¢=:}

= 0, then:

(the projection onto H of any representative of the germ A is a neighbourhood of zero).

Indeed, one may assume that M = en, H = {Zk+l = ... = Zn = O}, and N = {Zl = ... = Zk = O}, where k = dimH. Now, the right hand side of the above equivalence implies that N is a maximal subspace such that N nA = 0, and so codim A = dimN, i.e., dim A = k. Conversely, if dim A = k, then the germ A is k-regular (see the corollary in n° 4), which implies the right hand side of the equivalence (see n° 2). 6. Let !vI be a manifold. Then, for any analytic germs AI"'" Ak at a point a E M, we have the inequality

Indeed, one may assume that M is an n-dimensional vector space and

= O. Obviously, it is enough to prove that codim(A n B) S codim A + codim B for analytic germs A, B at O. We may assume that A i- 0 and B i- 0. First suppose that B is the germ of a subspace L. Clearly An L i- 0. a

Now the formula (#) from nO 5 shows that there is a subspace N such that codim(A n L) = dimN and An L n N = O. Hence, by the same formula, codim A ;:::: dim(L n N) and, since dimN S codimL + dim(L n N) (21), one derives codim(A n L) S codim L + codim A. In the general case we have h(A n B) = (A x B) n D, where D is the diagonal of M2 and h is eO) This inequality implies that dim A + dim B :

(VO is a submanifold of dimension k as a manifold).

6. Let V

c M

be a locally analytic subset, and let a E V.

THEOREM 2. If II, ... ,fT are representatives of generators of the ideal I(Va) and U is a sufficiently small open neighbourhood of the point a, then fi are holomorphic in U, we have V n U = {z E U: IICz) = ... = fT(Z) = O}, and

n T

(#)

Te V =

ker defi for

C

E VO

nU

.

PROOF. One may assume that M is a vector space and a = O. It is enough to show that there is an open neighbourhood U of a and II, ... ,fT satisfying the conditions in the conclusion of the theorem. Indeed, let g1, ... , g3 be representatives of generators of the ideal I(Va). If Ua C U is a sufficiently small open neighbourhood of a, then the functions fi, gi are holomorphic in Ua. Thus we have V n Ua = {gl = ... = g3 = O} n Ua (since the set {gl = ... = g3 = O} is a representative of the germ V (I(Va )) = Va; see II. 4.5) and fi

=

~j aijgj in Ua for some aij that are holomorphic in Ua

(since (ji)a E I(Va)). Therefore, for c E V O n Ua, we get the equality Te V = n~ ker dcg j ; the inclusion C follows because gi = 0 on V n Uo (see C. 3.11), whereas the inclusion :J is true because dc!i = ~j aij(c)d c 9j.

214

IV.2.7

IV. Geometry of analytic sets

Now, suppose that the germ Va is irreducible. Set k = dim Va. There exists a k-complete sequence O. Consequently, the set Zi is defined by the functions gil', where gi = (gil, ... ,gi.;), and so the set Z n Uis defined by the functions hI' = glVI ... gnv m , where v = (VI, ... ,Vm) E = {v: Vi = 1, ... ,si for i = 1, ... ,m} (see II. 3.1). In other words, Z n U = h-I(O), where h = (hal"" , haJ and e = {O'I,' .. , O'T}' Finally, for z E U, we have

e

rlh(z)1 ;::::

~ Ihv(z)1 = (~19Iv(Z)I) ... (~lgmv(z)l)

;::::

;:::: IgI(Z)I .. . 19m(Z)1 ;:::: em Q(z, ZI)P .. , Q(z, Zm)P ;:::: em Q(z,

zym .

§8. Analytically constructible sets Let M be an n-dimensional manifold. 1. By an analytically constructible leaf (in A1) we will mean a non-

empty connected submanifold r e M such that the sets rand r \ rare analytic (47). (47) Corollary 1 from proposition 5 in nO 3 justifies this terminology.

246

IV.8.2

IV. Geometry of analytic sets

Analytically constructible leaves are precisely the sets of the form V \ W, where V, Ware analytic sets, V is irreducible, and V* C W* V. Obviously, the condition W* V can be replaced by V cf- W. Indeed, if r is an analytically constructible leaf, then I' is irreducible (see 2.8, corollary 3 from proposition 2) and I'* c I'\r* I' (as r c I'0; see C. 3.7). Conversely, if V, Ware analytic sets, V is irreducible, and V* c W* V, then the set V \ W = VO \ (W n'V O ) is connected, open, and dense in VO (see II. 3.6; and 2.8, proposition 2). Hence it is a non-empty connected submanifold, V \

w=V

(see 2.1, proposition 1) and V \ W \ (V \ W)

= W.

If V, Ware analytic sets and V* c W, then the connected components of the set V \ Ware analytically constructible leaves. For every connected component H of the set V \ W is a submanifold (since it is an open subset of a connected component of the set yO) and the set jj is analytic (see 2.10, theorem 5), and so is the set jj \ H = jj n W. In particular, if V is an analytic set, then the connected components of the set VO are analytically constructible leaves. The Cartesian product of analytically constructible leaves is an analytically constructible leaf (48) . 2. Let A, B be families of subsets of the same space. We say that the family A is compatible with the family B if for every A E A and B E B either A c B or A C \B. In the case when B = {B}, we say that the family A is compatible with the set B. If, in addition, A = {A}, then we say that the set A is compatible with the set B. (Clearly, the family A is compatible with the family B if and only if each set A E A is compatible with each set B E B. Any reduction of sizes of the families preserves their compatibility.) If the family A covers the set B and is compatible with this set, then the set B is the union of some sets from the family A. A locally finite partition of the manifold Minto (disjoint) non-empty connected submanifolds r~ such that dim r~ = i and each of the sets ar; is the union of some of the sets r~, i < k, is called a complex stratification of the manifold M. Note that the last condition above (with the other ones satisfied) is fulfilled if and only if (1) the sets Ui9 r~ (k = 0, ... ,n) are closed, (2) the set r~ is compatible with the set r~ if i

< k,

§8. Analytically constructible sets

(3)

-k

rv nr~ = 0 for v f

fdk

= 0, ... ,n).

247

(49)

The Remmert-Stein theorem (see 6.3) implies the following PROPOSITION 1. The elements of any complex stratification of Mare analytically constructible leaves.

Indeed, let {r~} be a complex stratification of the manifold M. Obviously the sets r~ are analytically constructible leaves. Let 0 < k :::; n and assume that the r~ are analytically constructible leaves if i < k. Then the set V =

Ui n = dimM. If 2r > n, then

E = (Vo \ VI) U ... U (V2r - 2 \ V2r - l

) .

Indeed, by lemma 3, the condition is sufficient. Now suppose that the set E is analytically constructible. Then - in view of proposition 5 and the remark - the sets V; are analytic and V;+I is nowhere dense in V;, i = 0,1, .... Thus V; = 0 for i > n (see 2.5), and by applying lemma 3, we get the formula (** ). COROLLARY 1. If a set E c M is analytically constructible, then there exists an analytic set Z which is nowhere dense in E and such that E\Z C E.

(It is enough to take Z

=

Vd

COROLLARY 2. The analytically constructible sets are precisely the sets of the form where Vo :J ... :J V2 k+1 are analytic sets such that V;+I is nowhere dense in V, (i=0, ... ,2k).

COROLLARY 3. The class of all analytically constructible sets is the algebra of sets (52) generated by the class of all analytic sets. Proposition 7, together with formulae (*) and (**), implies: PROPOSITION 8. If {G,} is an open cover of the manifold M, then a set E C M is analytically constnlctible in M if and only if for each t the set EnG, is analytically constructible in G,. REMARK. It follows from proposition 8 that a set E C M is analytically constructible if and only if Ez E lC z for each z E M, where lC z denotes the (52) That is, it is a class of sets which is closed with respect to the operations of taking the union of two sets or taking the complement of a set.

252

IV.8.5

IV. Geometry of analytic sets

algebra of germs (53) (of sets at z) generated by the class of analytic germs at z. In other words, E is analytically constructible if it can be described locally by holomorphic functions (54), i.e., if each point of M has an open neighbourhood U such that E n U = Ui j Eij for some finite family of sets

n

Eij of the form {lij U.

= O}

or {lij

i- O}. Here hj

are holomorphic functions in

5. For each analytically constructible set E c M we define the sets Eo, E*, and E(k) (k = 0, ... , n) in the same fashion as that used for analytic sets (see 2.1). Thus we have the decomposition EO = E(O) u ... U E(n), and the set E( i) is a submanifold (of dimension i as a manifold) and is open in

EO (i = 0, ... , n). The space EO is locally connected and the sets E(i) are unions of its connected components. Each connected component of the set EO is a submanifold. We have also the formulae (E n G)O = EO n G, (E n G)* = E* n G, and (E n G)(k) = E(k) n G if G is an open set. The set E is said to be smooth if E = EO; then it is a locally analytic set. 4. If E is an analytically constructible set, then EO = (Va)O \ VI, Vi(E).

LEMMA

where

Vi =

In view of proposition 7, the sets Vi are analytic and we have the formula (**). Hence we have E \ VI = Vo \ VI, and so EO \ VI = (Vo)O \ VI. Therefore it is enough to show that EO n VI = 0. Now, if there was a point z E EO n VI, then, for some neighbourhood U of z, the set En U would be closed in U, and moreover, (VI \ V2 ) n U i- 0. Here we use the fact that z E VI and the set V2 is nowhere dense in VI. Since E C (Vo \ VdUV2 = E\(VI \ V2 ), we would have En U En U, contrary to the fact that En U is closed in U. PROOF.

*-

COROLLARY. In any analytically constructible set E, the set EO zs open and dense, while the set E* is closed and nowhere dense.

From lemma 4 and proposition 7, we have the following PROPOSITION 9. If a set E c M is analytically constructible, then so are the sets EO,E*, and E(k) (k = 0,,,. ,n). Furthermore, the connected components of the set EO are analytically constructible leaves.

Indeed, the set (Va)O is analytically constructible (see 2.4, theorem 1), and hence so is the set EO, as well as its connected components (see n° 3). This implies the analytical constructibility of the sets E(k), as well as the (53) That is, a class of germs (of sets at z) which is closed with respect to the operations of taking the union of two germs and the complement of a germ.

(54) See [27], p. 66.

§8. Analytically constructible sets

253

second part of the proposition (see corollary 1 from proposition 5). Let E c M be an analytically constructible set. We have dim E = dim E. (By 2.5, this is true for analytically COllstructible leaves. Then the general case follows by condition (2) in proposition 3; see II. 1.3 and n° 3.) It follows that dimz

E=

dimz E for any z EM.

If k = dimE ~ 0, then dim(E \ E(k») < dimE.

Indeed, consider a complex stratification of the manifold M which is compatible with E (see proposition 6). Then the sets E and E are unions of some leaves of dimension::; k of this stratification; they contain exactly the same k-dimensionalleaves - all of them are contained in E(k). Therefore the set E \ E( k) is contained in a union of leaves of dimension::; k - 1.

In particular, if f is an analytically constructible leaf, then dim of < dimf (55). Now let E C F be analytically constructible sets (in M). If E is nowhere dense in F and F =I 0, then dim E < dim F. (For E is nowhere dense in F; see 2.5.) In view of lemma 2, one derives the following cqui valence:

(E is nowhere dense in F)

-¢=?

(55) This follows also from the fact that

ar

(dim, E < dim. F for z E E) .

is nowhere dense in f' (see 2.5 and C. 3.7).

CHAPTER V

HOLOMORPHIC MAPPINGS

§1. Some properties of holomorphic mappings of manifolds Let M and N be complex manifolds. THEOREM 1. Let f: M --+ N be a holomorphic mappmg, and let V C M be a locally analytic set. Let kEN. If

rank z f :::; k

for

z EV ,

then f(V) is a countable union of submanifolds of dimension:::; k . Hence

dimf(V) :::; k .

Indeed, set m = k + dim V. The case when m = -00 is trivial. Suppose now that m 2 0 and the theorem is true if k + dim V < m. We have

V

= V*

U

UWo(i) U VI(i») ,

)

where Vo(i) = {z E V(i): rankzfF(i) = k} and VIC i) = {z E Veil : rankzfF(i) < k} (see IV. 2.1). Now V = 0 or dim V* < dim V (see IV. 2.4, theorem 1), and so the set f(V*) is a countable union of submanifolds of dimension:::; k. Next, if /.; = 0, then VIti) =

0; if k > 0, then, since

VICi)

is

an analytic subset (of dimension:::; dim V) of the manifold V(i) (see II. 3.7), i

the set f(V/ ») is a countable union of sub manifolds of dimension:::; k - 1.

§l.

255

Some propert.ies of holomorphic mappings

Finally, by the rank theorem (see C. 4.1), the set f(Vo(i)) is a countable union of submanifolds of dimension k. COROLLARY 1. We have dim f( E) ::; dim E for each analytically constructible set E C M. In partic-ular, dim feM) ::; dim M. The corollary holds because the statement is true when E is a submanifold. (See IV. 8.3, proposition 3.) COROLLARY 2 (SARD'S THEOREM). The set of critical values of the mapping f is a countable union of submanifolds of dimension < n = dim N, and hence it is a set of dimension < n.

e)

REMARK. It follows that the set is of measure zero and of first category e). THEOREM 2. Let U C M. If the graph of a mapping f: U --+ N zs locally analytic in AI x Nand dimx f 2: dim AI for x E f, then the set U zs open and the mapping f is holom07'Phic. PROOF. It is enough to show that if (a, b) E f, then the mapping f is defined and holomorphic in a neighbourhood of the point a. One may assume that M is a neighbourhood of zero in en, N = e k , and ( a, b) = O. Now, as dimf ::; n (see II. 1.4), the set f is of constant dimension n. Hence its germ fa is of constant dimension n (see IV. 3.1, proposition 4); since fa n (0 x N) = 0, it is n-regular (see III. 4.4). Consequently, by proposition 1 from IV. 1.4, it has a normal triple (D, Z, V) and one may assume that the crown V of the triple is a neighbourhood of zero in f (see IV. 1.2). But then the covering VO\Z --+ D \ Z is one-sheeted, which proves that V is the graph of a holomorphic mapping ill D (see IV. 1.8). Thus the mapping and holomorphic in D.

f

is defined

COROLLARY 1. If f: Af --+ N is a holomorphic injection and dim N ::; dimM, then the set f(M) is open in N and the mapping f: Ai --+ f(M) is biholomorphic.

REMARI{. If dim N > dim !vI, then f may not be an immersion. For example, the holomorphic mapping f: e 3 z --+ (Z2,Z3) E e 2 is injective, but daf = O. COROLLARY 2. Any holomorphic bijection f: !vI

--+

N is a biholomor-

e)

It is the image under! of the set. of crit.ical points of !, i.e., of points z E M for which the differential d,! is not surjective. Thus it is the image of the analytic set. {z EM: rank,! < n} (see II. 3.7).

e)

More generally, Sard's theorem says (see, e.g., [13]' Chapter XVI, §23; or [43], Chapter VII, §1) that the set of critical values of any Cco-mapping between smooth manifolds (with countable bases for topology) is of measure zero (i.e., its image under any chart is of measure zero). Hence, as an F,,-set, it is also of first category.

256

V.2.1

V. Holomorphic mappings

phic mapping. For then N = f(M), and so dim N :S dim M (see corollary 1 of theorem 1).

COROLLARY 3. Every local holomorphic homeomorphism of manifolds and, in particular, any holomorphic covering of manifolds is locally biholomorphic. COROLLARY 4. (THE ANALYTIC GRAPH THEOREM). Any continuous mapping f: IvI ----7 N whose graph is analytic in M X N is holomorphic. In fact, take (a, b) E F. The sets ff!, where n is any open neighbourhood of the point a, form a base of neighbourhoods of the point ( a, b) in f. Therefore, since dim ff! 2: dim M (by corollary 1 of theorem 1 applied to the natural projection n X N ----7 n), we have dim(a,b) f 2: dimM. REMARK. Instead of continuity, it is enough to assume local boundedness of the mapping f. For the latter property implies continuity, as f is a closed set in M X N (see B. 2.3). In particular, if the manifold N is compact, the assumption that the mapping f is continuous is redundant. In the general case however, it is necessary. For instance, the graph of the mapping f: C ----7 C given by f(O) = 0 and fez) = Z-1 for z I- 0, is analytic in C 2 , but the mapping f is not even continuous.

§2. The multiplicity theorem. Rouche's theorem Let M and N be manifolds of the same dimension n > 0, and let f: M ----7 N be a holomorphic mapping. 1. We say that the mapping f is light at the point a E M if a is an isolated

point of its fibre e) f- 1 (J(a)). In such a case, we define the multiplicity of the mapping f at the point a by

where nand 6. are sufficiently small neighbourhoods of the points a and f(a), respectively (so that the right hand side is independent of nand 6.). Furthermore, if the neighbourhood n is sufficiently small, the equality (*) is true for each neighbourhood 6.. Since, denoting the right hand side by

e)

By the fibre of the point a we mean the fibre of (See C. 4.1, footnote (19)).

I

containing a, i.e., the set

1-1 (I( a)).

257

§2. The multiplicity theorem

m(fl, ~), we have maf = m(fl,~) if fl C flo, it follows that ~ C ~o for some neighbourhoods flo and ~o. One may assume that f(flo) C ~o. Then, for fl C flo and for any ~, we have m(fl,~) = m(fl, ~ n ~o) = maf.

H G is an open neighbourhood of the point a E M and the image f( G) is contained in an open set HeN, then the mapping f is light at a if and only if the mapping fa: G - - t H is light at a, and we have mafa = maf. H N is a holomorphic mapping eO). Suppose that for each t E H the set ft- 1(0) is finite and ft(z) =1= 0 outside a compact set E c M (independent of t). Then the function

is locally constant. If the manifold M is biholomorphic to an open set in a vector space, then the finiteness of the sets f t- 1(0) follows from the other assumptions (11). First, we will prove - under the assumptions of Rouche's lemma - the following LEMMA

2. Let w be a regular value of the mapping ft o ' Assnme that

the set ft~I(W) is finite, and that ft(z) 1

# ft- (w)

= # ft~l (w)

=1=

w in M \ E for t E H. Then

for each t from a neighbonrhood of the point to.

Indeed, we have ft~I(11') = {a1, ... ,ad. Therefore F(to,a;) = w, 1, ... , k, and, by the implicit function theorem, there are mutually disjoint open neighbourhoods D 1 , ... ,D k of the points aI, ... , ak and a neighbourhood U of the point to such that if t E U, then F(t,b;(t)) = w for a unique

b;(t) E Di (i = 1, ... , k). Now, fto(z) =1= w in the compact set E \ U D;; hence ft (z) of w in E\U Di for t from a neighbourhood Uo C U of the point to. Thus, for t E Uo , we have ft- 1 ( 11') = {b 1 (t), . .. ,b,,(t)}, and so

# ft- 1 (w)

=

# ft~l (w).

PROOF of Rouche's lemma. It is enough to show that in the case when H is an open neighbourhood of zero in a normed vector space L, the function t ---> v(Jt) is constant in some neighbourhood of zero. We may assume (by taking a larger E, and a smaller l'vf and H) that for some E > 0 (and after

(9) Since!(z)7'=winM\U12j.

eO) Holomorphic dependence on the parameter t is not essential. See the remarks following RouclH~'s

theorem below.

(11) See the proposition in IV. 5.

262

V.2.3

V. Holomorphic mappings

furnishing N with a norm), we have IF(t, z)1 ;:;:: c: in H x (M \ E), and B = {It I c:} c H. Now, consider the holomorphic mapping CP: H x M ---> Lx N defined by cp(t,z) = (t,F(t,z)). Then

:s

cp-l(t,W) = t x It-1(w) for t E H, wEN, the set cp-l(O) is finite, and we have Icp(t,z)1 ;:;:: c: in (H x M) \ (B x E). Therefore, according to lemma 1, there are connected open neighbourhoods U CHand ~ C {Iwl < c:} of the zeros in Land N, and a nowhere dense analytic subset A in U x~, such that #cp-;l(s,w) = v(cp), i.e.,

#Is-1(w) = v(cp) if (s,w) E (U x~) \ A .

(#)

Let t E U. In view of lemma 1, there is an open neighbourhood ~o C of zero in N and a nowhere dense analytic subset Wo of ~o such that

(##)

v(Jt) = #It-1(w) and } w is a regular value of It

provided that w E

~

~o \ Woo

Now, the set {w E ~: A W is nowhere dense in U}, where A W = {t E U : E A}, is dense in ~. For if it were not, then, due to connectedness of U, we would have A :J U x ~' for some non-empty open ~' (see II. 3.6). Hence there exists a point Wo E ~o \ Wo such that AWo is nowhere dense in U. Note that Is(z) -::J Wo in M \ E for s E H (as F(s, z) ~ ~o when (s, z) E H x (M \ E)). Because of this and (##), we can now apply lemma 2. Thus # I;l( wo) = # It-l ( wo) for some s E U\A WO. Then (s, wo) E (Ux ~)\A. In view of (#) and (##), it follows that

(t, w)

So we have proved that the function t

--->

v(Jt} is constant in U.

REMARK. Using Rouche's lemma, one can deduce the corollary from n° 2 as follows. One may assume that M, N, L are open neighbourhoods of zero in a normed vector space and that a = b = 1/;( a, b) = O. By the implicit function theorem (see C. l.13), there is an c: > 0 such that {Izl < c:} c M, {Iwl < c} eN, and

(Izl < c:, Iwl < c:, 1/;(z,w)

= 0)

=?

w

= O.

J(z) = 0) =? z = 0 tEe and It I < 2, then

Furthermore, there exists 0 < 5 < tc: such that (Izl < 5, and

J({lzl < 5}) c {Iwl < c}.

It follows that if

263

§2. The multiplicity theorem

the only zero of the functiongt(z)

=

'ljJ(tz, fez)) in {izi

-00

view of Rouche's lemma (and because go = f, where biholomorphic at 0; see C. 3.13), we conclude that mog v(go) = mogo = mof·

< 5} is O. Thus, in

-0:

z

---t

'ljJ(0, z) is

= mogl = V(gl)

=

ROUCHE'S THEOREM. Assume that the manifold !vI i.'3 biholomorphic to an open subset of a vector space. If g: M ---t N is a holomorphic mapping such that (after endowing N with a norm) the following inequality holds Ig(z)1


O. Therefore one can apply Rouche's lemma to the mapping ft(z) = f(z) + tg(z) defined in {It I < r} X M. So the function t ---t v(ft} is constant in the disc {It I < 1'} (because the latter is connected). Thus v(f + g) = V(fl) = v(fo) = v(f). REMARKS. In the case of an arbitrary manifold }'I, the following version of Rouche's theorem is true: If the set f- 1 (0) is finite and contained in the interior of a compact set E C M, then there is 5 > 0 such that a holomorphic mapping g: M ---t N satisfying the conditions: Ig(z)- f(z)1 < 5 in E and g(z) f. 0 in M\E implies that the set g-I(O) is finite and v(g) = v(f).

In fact, let M; be neighbourhoods of the zeros of f such that Mi C E, the closures of the Mi'S are mutually disjoint, and each },Ii is biholomorphic to an open ball in en. It is enough to apply Rouche's theorem to the restrictions fM. and to note that inf{lf(z)1 : z E E \ U M;} > O. It is now easy to see that in Rouche's lemma the holomorphic dependence on the parameter t is not essential. It suffices to assume that F( t, z) is a continuous mapping in H x }'I which is holomorphic with respect to t, where H is a topological space (the rest of the assumptions remain unchanged). (12) Obviously, in such a case, the sets 1-1(0) and (f proposition in IV. 5.)

+ g)-leO)

are finite. (See the

264

V.3.1

V. Holomorphic mappings

§3. Holomorphic mappings of analytic sets L Let V and W be locally analytic subsets of manifolds M and N, respecti vely.

We say that a mapping I: V ---+ W is holomorphic if each point of V has an open neighbourhood U in M such that Ivnu is the restriction of a holomorphic mapping of U into N. In the case when V and Ware submanifolds, the above notion coincides with that of a holomorphic mapping of manifolds (see C. 3.8). Clearly, if M' C M and N' C N are submanifolds that contain V and

W, respectively, then for the mapping I: V ---+ W to be holomorphic it is irrelevant whether V, Ware regarded as locally analytic subsets of the manifolds M, N or of the manifolds M', N' (see II. 3.4). Combining the above definition with basic properties of (locally) analytic sets, analytically constructible sets, and holomorphic mappings of manifolds, one deduces the following properties: Obviously, every holomorphic mapping between locally analytic sets is continuous. If V is the union of a family of open sets V, in V, then a mapping I: V ---+ W is holomorphic if and only if all the restrictions Iv. : ---+ W are holomorphic. The restriction of a holomorphic mapping I: V ---+ W to a locally analytic subset Z C V of M is holomorphic. If W' C W is a locally analytic subset of N, then a mapping I: V ---+ W' is holomorphic if and only if I: V ---+ W is holomorphic. The graph of a holomorphic mapping I: V ---+ W is locally analytic in M X N; the inverse image under a holomorphic mapping I: V ---+ W of a locally analytic subset T C W of N is locally analytic in M. The natural projections V X W ---+ V and V X W ---+ Ware holomorphic.

v..

The composition of holomorphic mappings is holomorphic. The Cartesian product and the diagonal product of holomorphic mappings are holomorphic. Note also that if g: L ---+ M is a surjective submersion, then

(J: V

---+

W is holomorphic) ~ (J

0

g: g-l(V)

---+

W is holomorphic) .

The above follows directly from the definition of submersion (just as in C.4.2). Suppose now that V, Ware analytic subsets of M and N, respectively, and that I: V ---+ W is a holomorphic mapping. Then the graph of the mapping I is analytic in M x N. The inverse image of an analytic set or an analytically constructible set T C W in N is - respectively - analytic or

265

§3. Holomorphic mappings

analytically constructible in M. If the image feZ) of an analytic set Z C V in M is analytic in N; then, if the set Z is irreducible, so is its image feZ). We have the inequality dimf(E) ~ dimE for each analytically constructible set E

c

V in lvi.

2. Let f: V ~ W be a holomorphic mapping of locally analytic sets V, W in the manifolds M, N, respectively. If z E Vo, then some open neighbourhood n in V of the point z is a submanifold (of the manifold M), and then rankzf is well-defined as the rank at the point z of the mapping fn : n ~ N. Naturally, in the case when V and Ware submanifolds, the above definition coincides with that given in C. 3. 12. THEOREM 1. If the set V is analytic in ]11, then for any connected component W of the set VO and for any kEN, the set

{z

E W: rankzf ~ k}

is analytically constructible in M.

PROOF. According to theorem 4 from IV. 2.9, we have tV where Vi is a simple component of V. Hence

= 11;0 \

V*,

{z E W: rank,f ~ k} = {z E 11;0: rankzfv; ~ k} \ V*. Therefore one may assume that V is of constant dimension, and it is enough to prove that the set

E= {z E Vo: rankzf ~ k} is analytically constructible in ]V[ (see IV. 2.8, corollary 1 from proposition 2). Since the property of being analytically constructible is local (see IV. 8.4, proposition 8), one may assume that N = en, f = Fv, where F = (F1 , ... , Fn): M ~ en is a holomorphic mapping. Moreover, it is enough to show that each point a E V has an open neighbourhood U in M such that the set En U is analytically constructible in U. Now, theorem 2 from IV. 2.6 says that there is an open neighbourhood U of the point a and holomorphic functions G 1 , , , . , G r on U such that Tz VO = n~ ker dzG; for z E VO n U. Then, by putting G = (G 1 , ... ,G r ), we have ker dzfvo = ker dz(F, G) for z E VO n U (see C. 3.11). It follows that

Enu

= {z

E U: rank.(F,G) ~ k+codim V}nVO,

266

V.3.2

V. Holomorphic mappings

which shows that the set En U is analytically constructible in U (see II. 3.7; and IV. 2.4, theorem 1). THEOREM 2. Let kEN . We have the inequalities

+ dimf(V)

if

dimf-l(w);:: k

for

wE f(V);

dimV:::;k+dimf(V)

if

dimf-l(w):::;k

for

wEf(V).

dim V;:: k

PROOF of the first inequality (13). One may assume that N =I- 0. Next, we may assume that W is an affine space and that the set f(V) is open in W. Indeed, there is a submanifold r C f(V) of dimension equal to dim f(V) which is biholomorphic to an open subset of an affine space; then it suffices to have the inequality for the mapping ff-'(r): f-l(r) ~ r (because

Uf-'(r))-I(W) = f-l(w) for w E r). Clearly, one may assume that V is analytic. Let l = dim W. The case when 1= 0, is trivial. Suppose that 1 > 0 and the inequality is true for any (1 - 1 )-dimensional space W. Take a nonconstant affine mapping X: W ~ C. Since the set (X 0 f)(V) is open and non-empty, it contains a point c which is not the value of any constant restriction of X 0 f to a simple component of V. Then Wo = X-I (c) is an affine space of dimension I-I, while Vo = f-I(WO) = (X 0 f)-l(c) is a non-empty analytic set of dimension < dim V (see IV. 2.8, proposition 3). Consider now the mapping fvo : Va ~ Wo. Since the set f(Vo ) = f(V) n Wo is non-empty and open in Wo , and fVol(W) = f-l(w) for w E f(Va), we have dim V> dim Vo ::::: k and so dim V;:: k

+ dimf(Vo ) = k + dimf(V)

- 1,

+ dimf(V).

PROOF of the second inequality. Consider the natural projection 7r: f ~ N. We have 7r(J) = f(V) and 7r- I (w) = f-l(w) X w. Hence dim7r- l (w) :::; k for w E 7r(J). Thus dim V :::; dimf :::; k + dim7r(J) = k + dim f(V) (see n° 1 and II. 1.4). COROLLARY 1. If f is a mapping whose fibres are discrete, and in particular, if f is an injection, then dim f(V) = dim V. If, in addition, V is of constant dimension, then so is f(V).

COROLLARY 2. Let Z C W be a locally analytic subset of N. If dimf-l(w) = k for each w E Z, then dimf-l(Z) = k + dimZ. Moreover, if the mapping f is open and both the subset Z and all the fibres f- l (w), (13) Based on an idea due to Narasimhan [33].

267

§3. Holomorphic mappings

w E

Z are of constant dimension, then the subset f- I (Z) is also of constant J

dimension (14).

Indeed, to show the second part of the corollary, let T = f- I (Z), and let D be an arbitrary open neighbourhood in V of any point of T. For each

w E f(T n D), the dimension of the fibre (fTnn) -1 (w) = f- I (w) n D is equal to k and the dimension of the set f(T n D) = Z n f(D) is equal to dim Z. Therefore dime T

n D)

= k

+ dim Z.

For each z E V, denote by lzf the germ at z of the fibre of z:

THEOREM 3 (semiconti~uity). The function

V '3 z ---. dim lz! ~s upper semicontin1LOus: 101' each a E V we have the inequality dim l z f < dim laf in a neighbourhood (in V) of a.

PROOF. One may assume that M and N are vector spaces. Set n = dimM. Fix an arbitrary kEN. Now, for any point z E V, the condition diml z l :s; k is equivalent to the condition: for some (n - k)-dimensional subspace L eM, the point z is an isolated point ofthe set 1- 1 (f( z)) n (z + L) (see III. 4.5, formula (#)). The latter occurs if and only if (z,f(z)) is an isolated point of the set

(rl (f(z)) n (z + L))

X

fez)

=

f n ((z,l(z)) + (L

X

0))

Now, by the corollary from proposition 1 in IV. 1.4, if our condition is satisfied for the point z = a E V, then it is also satisfied for each point z from some neighbourhood (in V) of the point a. In view of theorem 2, we get the following COROLLARY. We have dim lzf ~ dimz V - dimf(V) fOT z E V. Indeed, if it were not true, then, denoting the right hand side of the inequality by p, we would have an open neighbourhood U (in V) of the point (14) The assumption that the mapping I is open cannot be omitted even in the case when V and Ware submanifolds. For instance, consider the mapping I: C3 3 (t, z, w) --+ (tw, zw, t) E C 3 and the set Z = X X C of constant dimension 1. All the fibres 1-1(0) = (0 X C x 0) U (0 x x C) and 1- 1 (0,0,c) = c x C x 0, where c =I 0, are of constant dimension 1, whereas the variety 1-1 (Z) (C 2 X 0) U (0 x x C) is not of constant dimension.

°

° °

=

°

268

V. Holomorphic mappings

V.3.3

z such that dimfu 1 (w) :S p - 1 for W E f(U), and dimz V = dimU < p - 1 + dimf(U) < p + dimf(V). This contradicts the definition of p. We define the rank of the mapping f by the formula

Clearly, this is an extension of the previously given definition in the case when V and Ware submanifolds (see C. 3.12).

If the set V is irreducible, then rank open subset n of V; next, dim I z f

In

= rank

2 dim V - rank f for

f for any non-empty

z EV ,

and hence

dimf-l(w) 2 dim V - rank f for wE f(V) (cf. nO 3 below)

e

6 ).

Indeed, the set G = {z E Vo: rank z f = rank f} is open and dense in the submanifold VO (see II. 3.7; and IV 2.8, corollary 1 from proposition 7 2). In view of the rank theorem (see C. 4.1), we have inequality (*) ) for z E G, and hence also for z E V, because of the semicontinuity theorem (theorem 3).

e

"\VHITNEY'S

LEMMA. We have rank fv. :S rank f·

rank fv'. We may assume that V* i= 0, and then i= 0 which is open in (V*)O and is a submanifold (see C. 3.12). Set k = dim6. According to the rank theorem (see C. 4.1), we may assume that r = f(6) is an r-dimensional submanifold, and that the fibres of the mapping f c,.: 6 ---t rare (k - r )-dimensional submanifolds. Hence dim lzfv' = k - r for z E 6. Now, if we had rank f < r, i.e., PROOF.

Let r

=

rankzfv· = r in some set 6

rank z f < r for z E VO , then we would have dimf(VO) < r (see §1, theorem 1), and so r 1. f(VO). Thus, there would exist a point a E 6 such that f( a) rf. f(VO). This would mean that f- 1 (J( a)) C V*, and hence we would have dim laf

=k- r .

e If V = 0, we put: rank f = e Clearly, without the assumption of irreducibility of V, the inequality is no longer true. e7) In fact, we have equality. S

6

) )

-ex).

269

§3. Holomorphic mappings

It follows from the semicontinuity theorem (theorem 3) that, for some open neighbourhood U of the point a, we have

dim lz! ::; dim laf if z E U . Set m = dim Va. As a consequence of the decomposition into components of constant dimension (see IV. 2.9), there is a point c E vern) n U. In addition, it can be chosen so that in a neighbourhood ofthe point, rankzfv(m) is constant (see C. 3.12) and hence equal to some s < T. Therefore, by the rank theorem (and because Ic!V(m) = Ie!), we would have m = s

+ dim lef < r + dim laf = k

This is impossible, because dim Va THEOREM 4. rank f

~

.

dim /:).a = dim /:)..

= dim f(V).

Indeed, the inequality dimf(V) ~ rank f follows from the fact that fey) contains (except for the trivial case V = 0) non-empty submanifolds of dimension rank f. (See C. 3.12 and the rank theorem in C. 4.1.) Set k = dim V. The opposite inequality is trivial when k = -CXJ. Assume now that k ~ 0 and the inequality is true if dim V < k. Then - in view of theorem 1 from IV. 2.4 and Whitney's lemma - we have dim f(V*) ::; rank fv' ::; rank f. But also dimf(VO) ::; rank f (see §1, theorem 1). Therefore dimf(V) < rank f. COROLLARY 1. If Z

c

V is a locally analytic set (in M), then

rank

fz ::; rank f .

COROLLARY 2. If the set V is irred1lcible, then the set fey) is of constant dimension (18). In fact, for any open neighbourhood 6 in W of any point of the set f(V), since f(V)n/:). = it-I(,:,) (J-I (/:).)) , we have dim(J(V)n/:).) = rank iF-I(':') = rank f. 3. Let f: V --+ W be a holomorphic mapping between analytic subsets V, W of complex manifolds M, N, respectively, and aSS1lme that the set V is iTred1lcible. (Thus VO is a connected submanifold and rank fn = rank f for every open set n i- 0 in V.)

(18) Obviously, the assumption of irreducibility of V cannot be replaced by the assumption that V is of constant dimension.

270

V.3.3

V. Holomorphic mappings

We define the generic dimension of the fibres of the mapping f by the formula

>..(1) = min{dimlzf:

Z

E V} .

(If V = 0, we put >..(1) = -=.) Owing to the semicontinuity theorem (theorem 3 in nO 2), the set V in the equality (*) can be replaced by any dense subset of V. Clearly, dimf-l(w) ;::: >..(1) for wE f(V) . Consider the set

C(1) = {z E Vo: rankzf < rank j} U V* . It is a nowhere dense subset of V (see II. 3.7; and IV. 2.1, proposition 1), analytic in M (see n° 2, theorem 1; and IV. 8.3, proposition 5). We have the equalities

(1)

dim V

= >..(1) + rank f

and

(2)

dimlz!

= >..(1)

for

Z

E V \ C(1) .

For, according to the rank theorem (see C. 4.1), we have dim V = dimlzf rank f for Z E V \ C(1).

+

We also have

dimf-l(w) = >..(1) in a dense subset of the set f(V) .

(3)

For otherwise one would have dim f- 1 (w) > >..(1) in an open set ,6 i=f(,6). Hence, in view of theorems 2 and 4 from nO 2, we would have dim V;::: dimf-l(,6) > >..(1)

+ rank

f/-1(f',.) = >..(1)

+ rank

0 in

f ,

in contradiction with equality (1). Moreover, if V # 0, then equality (3) is true in the set f(V), except for an F(7-subset of f(V) of dimension less than dimf(V). Indeed, let k > 0, and suppose that the above is true when dim V dim V 0 is trivial; see III. 4.3). Now let dim V = k. Set

=

z=

{z E VO : rank, f

< rank n

.


A(J) and set C = C(J). Since dim C < dim V (see IV. 2.5), it is enough to show that Zk(J) = Zk(Je). The inclusion :J is trivial. Now let z E Zk(J), i.e., dim l,f ~ k. By (2), we must have z E C. Consider the fibre

r = f- 1 (I( z)).

In view ofthe rank theorem (see C. 4.1) and (2), the set

I' =

r \ C is a A(J)-dimensional submanifold, and so dim I' z :s; A(J) < dim Izf. As r = (r n C) U I' and r n C = fc/ (Ic(z)) , we have Izf = rz = Ide U I'z, and so dim Ide = dim lzf

~

k (see II. 1.6). Consequently, z E Zk(Je).

Using Remmert's proper mapping theorem (see 5.1 below) we obtain the following corollary 9) :

e

COROLLARY 1. If the mapping f is proper, the set IV is irreducible, and dim V ~ dim W, then the set {tv E TV: # P (tv) < CXJ} is open and dense in IV, and its complement in TV is ana.lytic (and nowhere dense). (19) This corollary will not be used until Remmert's theorem is proved.

272

V. Holomorphic mappings

V.3.4

Indeed, since the fibres f- 1 (w) are compact, the complement of the above set in W is the set Z = {w E W: dimf-l(w);::: I} (see III. 4.3). The latter is analytic, because it is the image of the analytic set {z E V : dim I z f ;::: I}. It is nowhere dense in W, for otherwise we would have Z = W :j:. 0 (see IV. 2.8, proposition 3), and consequently, dim V ;::: 1 + dim W (see nO 2, theorem 2), contrary to our assumption. COROLLARY 2. If M is an n-dimensional vector space and A is an analytic k-dimensional germ at 0 E M, then the set of coordinate systems cp E Lo (M, en) in which the germ A is k-regular is the complement of a nowhere dense algebraic subset of L(M, en) (and hence it is open and dense in Lo(M, en); see II. 3.2).

In fact, it is sufficient to show that the complement Z of our set in

L(M, en) is algebraic (see III. 2.7). Let V be an analytic representative of the germ A in an open neighbourhood U of zero. One may assume that k ;::: 0, i.e., that 0 E V. Let N = {ZI = ... = Zk = O} c en. A coordinate system cp E Lo(M, en) belongs to our set if and only if cp(A) n N = 0 (see III.

n cp-l(N))o = O. l(o,cp)7r = (V n cp-l(N))o

4.4) or, equivalently, if (V

It follows that the set Z is a

cone. Now we have

x cp for cp E L(M, en), where

7r: A

----t

L( M, en) is the natural projection of the set 11.= {(z,cp) E U x L(M,e n ): z E V,cp(z) E N} .

The latter is analytic in U x L(M, en). Thus

Z

=

{cp E L(M, en): diml(o,cp)7r > O} U 2: ,

where Ij = L(M, en) \ Lo(M, en) is an algebraic set (see II. 3.2). Therefore, in view of theorem 5, the set Z is analytic, and so it is algebraic, according to the Cartan-Remmert-Stein lemma in II. 3.3. Corollary 2 implies that the sets of coordinate systems in Lo(e k , e k ) and Lo(M, en), described in the proposition in III. 2.7 and its corollary (see footnotes eO) and eOa) in IlL3), are the complements of nowhere dense algebraic sets in L(e k , e k ) and L(M, en), respectively. (Hence they are open and dense.) 4. Let V and W be locally analytic subsets of the manifolds M and N, respectively. A mapping f: V ----t W is said to be biholomorphic if it is bijective and the mappings f, f- 1 are holomorphic. (In the case when V and Ware submanifolds, the definition coincides with that given before; see C. 3.10).

273

§3. Holomorphic mappings

Such a mapping is a homeomorphism. If there is a biholomorphic mapping f: V ---7 lV, we say that the subsets l'V and V are biholomorphic. Obviously, the inverse of a biholomorphic mapping is biholomorphic and so is the composition of biholomorphic mappings. If f: V ---7 W is a biholomorphic mapping, then if Z c V is a locally analytic set in AI, the set feZ) is locally analytic in N and the mapping fz: Z --+ feZ) is biholomorphic. Note also that if g: V ---7 W is a holomorphic mapping, then the natural projection from the graph of 9 to V is biholomorphic. Let f: V

---7

W be a biholomorphic mapping.

Biholomorphic mappings "preserve" regular points (and their dimensions) as well as singular points: f(V(k)) = W(k), and hence f(V*) = W*. Indeed, let a E V(k) and b = f(a). The mapping f- 1 is the restriction of a holomorphic mapping F of a neighbourhood of b in N in the manifold M. Therefore F(J(z)) = z in a neighbourhood of a in V. Since fV(k) : V(k) --+ N is a holomorphic mapping of complex manifolds, we have (dbF) (dafv(k) ('U)) = 'U in Ta V, and so the differential dafv(k) is injective. Hence (see C. 3.14) the image under f of a neighbourhood in V of a, i.e., a neighbourhood in W of b, is a k-dimensional submanifold of the manifold N. This means that bEW(k).

In particular, if V is a submanifold, then so is W. It follows that every biholomorphic mapping preserves dimension:

dimf(E) = climE for E

c

V .

Now assume that V, vV are analytic (in AI and N, respectively). Any biholomorphic mapping between analytic sets preserves analyticity and analytic constructibility. That is, if E c V is an analytic (or analytically constructible) set in .!vI, then f( E) is analytic (or analytically constructible) in N. Any biholomorphic mapping (of analytic sets) preserves irreducibility of analytic subsets. This means that if a set Z C V is an irreducible analytic subset of M, then its image feZ) is an irreducible analytic subset of N. Therefore any biholomorphic mappings between analytic sets preserves the decomposition of an analytic subset into simple components as well as the decomposition into components of constant dimension. We say that a mapping f: V ---7 1¥ is biholomorphic at a point a E V, if there are open neighbourhoods 6" of a in V and n of f( a) in W such that the mapping f i',.: 6" ---7 n is biholomorphic. If this is the case for each a E V, we say that f is a locally biholomorphic mapping eO). eO) These definitions generalize those from C. 3.10.

274

V. Holomorphic mappings

V.4.1

(Of course, a locally biholomorphic mapping is holomorphic and is a local homeomorphism.) Note that if 1: V - - t W is biholomorphic at a point a, then a E VO ~ l(a) E Woo PROPOSITION. Suppose that 1: V - - t W is a holomorphic bijection and W is a submanilold of N. Then the following conditions are equivalent:

(1) V is a submanifold of M; (2) V is of constant dimension;

(3)

1 is

a homeomorphism;

If any of the conditions is satisfied, then f is biholomorphic

(21).

One may assume that W = N. The implications (1) ===} (3) (with the last conclusion) and (3) ===} (2) follow from corollary 2 of theorem 2 in §1 and corollary 2 of theorem 2 in n° 2, respectively. Now suppose that (2) is satisfied. Then 1 is of constant dimension n = dimN, because the natural projection 1 - - t V is biholomorphic and the natural projection 1 - - t N is bijective (see n° 2, corollary 1 from theorem 2). Take an arbitrary (b, a) E 1-1 C N X M. It is enough to show that some neighbourhood of (b, a) in f- 1 is the graph of a holomorphic mapping of an open set in N. Indeed, it would then follow that the mapping 1-1 is holomorphic (since it would be biholomorphic at each bEN); hence 1 would be biholomorphic and V would be a submanifold. Now, the germ (J-l )(a,b) is of constant dimension n. One may assume that M = em, N = en, and (b, a) = 0. As (J-l)O n (0 X M) = 0, the germ (J-l)o is n-regular (see III. 4.4), and hence it has a normal triple of dimension n whose crown is a neighbourhood of the point in f- 1 (see IV. 1.4, proposition 1; and IV. 1.2). But the multiplicity of this triple must be 1, and so (see IV. 1.8) its crown is the graph of a holomorphic mapping (on an open neighbourhood of the origin). PROOF.

°

The conditions (1 )-(3) are essential. A counter-example is the holomorphic bijection e 2 :J {zw = 1} U (0,0) 3 (z,w) - - t Z E e . COROLLARY. If a holomorphic mapping f: V - - t VV is a local homeomorphism and W is a submanifold of N, then V is a submanifold of M and f is locally biholomorphic.

(21) Note that if f: V --+ W is a holomorphic bijection and V is a submanifold of M, then W need not to be a submanifold even if f is a homeomorphism and V, Ware irreducible (see IV. 2.1, footnote (10)).

275

§4. Analytic spaces

§4. Analytic spaces Complex manifolds are locally modelled on open sets of the space en. Similarly, analytic spaces are locally modelled on analytic subvarieties. Now we are going to define analytic spaces and state their elementary properties (in n° 1-3), which can be checked easily, usually in the same way as for manifolds.

1. By an analytic space we mean a topological Hausdorff space X with an analytic atlas (in the restricted sense), i.e., with a family of homeomorphisms are charts on the spaces X and Y, respectively, then 'fi x 1/> is a chart on the space X x Y.

2. Every complex manifold is an analytic space; it has the induced structure of an analytic space defined (in a unique fashion) by any complex atlas of the manifold. We say that an analytic space X is an n-dimensional manifold if its structure is induced by the structure of an n-dimensional manifold (the latter must then be unique; see below). Note that an analytic atlas that consists of charts whose ranges are open subsets of en must be a complex atlas (see 3.4). It follows that: The structure of an analytic space can be induced by at most one structure of an n-dimensional manifold. An analytic space X is an n-dimensional manifold if and only if it has an atlas that consists of charts onto open subsets of en (and this atlas is then the complex manifold atlas inducing the structure of an analytic space on X). Let X be an analytic space. If it is an n-dimensional manifold, then so is each of its open subsets (with the induced structure of an analytic space). If {G,} is an open cover of the space X, then the latter is an n-dimensional manifold if and only if each of the analytic spaces G, is an n-dimensional manifold.

§4. Analytic spaces

277

An analytic space X with an atlas {cp,: G, ~ Y,.} is an n-dimensional manifold if and only if all Y,. are n-dimensional submanifolds. (As for the sufficiency of the condition, it is enough to take the atlas {1{>'K 0 cp,}, where {1{>'K} are atlases ofthe manifolds 1f" respectively. The condition is necessary, because every biholomorphic mapping of analytic sets preserves regular points and their dimension; see 3.4.) In particular, a locally analytic subset of a manifold M, regarded as an analytic space, is an n-dimensional manifold precisely when it is an ndimensional submanifold of the manifold M. 3. A subset Z of an analytic space X is said to be analytic or locally analytic if its image under every chart on an analytic subset of a manifold is analytic or locally analytic, respectively, in this manifold. It is enough if this is true for each chart of a given atlas {cp,: G, ~ Y,.}. Then the set Z has the induced structure of an analytic space, defined (in a unique way) by the atlas of the restrictions {(cp,)z: Z n G, ~ cp,(Z)}. The set Z furnished with this structure is called an analytic subspace of the space X. A subset of an analytic space is analytic if and only if it is locally analytic and closed. Clearly, every open subset G of an analytic space X is locally analytic (25); note that the charts of the analytic space G are precisely the charts of the space X whose domains are contained in G. The locally analytic subsets of the analytic space X are precisely the analytic sets in the open subsets of X (regarded as analytic spaces). If Z is an analytic suo space of the analytic space X, then a set W C Z is locally analytic in Z if and only if it is locally analytic in X; then X and Z induce the same structure of an analytic space on W. In the case when Z is analytic (i.e., if, in addition, Z is closed), then a set W C Z is analytic in Z if and only if it is analytic in X. If Z, Ware (locally) analytic subsets of analytic spaces X, Y, respectively, then Z X TV is a (locally) analytic subset of the space X X Y. Then the Cartesian product Z X W of the analytic spaces Z and VV coincides with the analytic subspace Z X TV of the analytic space X X Y. The union of a locally finite family and the intersection of a finite family of analytic subsets of an analytic space are analytic. If G is an open subset and Z is a (locally) analytic subset of the analytic space X, then Z n G is a (locally) analytic subset of the analytic space G. If {G,} is an open cover of the analytic space X, then a set Z C X is (locally) analytic if and only if each of the sets Z n G, is (locally) analytic in the analytic space G" respectively.

If an analytic space X is a manifold, then the notions of an analytic and a locally analytic subset defined above coincide with those defined in II. 3.4. (25) Obviously, the induced structure 011 G defined here coincides with that defined in nO l.

278

V.4.4

V. Holomorphic mappings

If an analytic subspace Z of the analytic space X is an n-dimensional manifold, then the set (the analytic subspace) Z is called an n-dimensional submanifold of the analytic space X. Any k-dimensional submanifold (and in particular, any open subset) of an n-dimensional submanifold of an analytic space X, is also a k-dimensional (respectively, n-dimensional) submanifold of the space. If a set Z C X is the union of a family of its open subsets, each of which is an n-dimensional submanifold, then the set Z is an n-dimensional submanifold. If {cp,: G, - - t V,} is an atlas of the analytic space X, where V, are analytic subsets of manifolds M" respectively, then a set Z C X is an n-dimensional submanifold if and only if all cp,(Z) are n-dimensional submanifolds of M" respectively. 4. Let X be an analytic space, E eX, and a EX. The dimension of the set E at the point a is well-defined by the formula dima E = dim'P(a) cp(E), where cp is any chart whose domain contains a. Next, we define dim E = sUPzEX dimz E (and we put dim 0 = -(0). Clearly, any non-empty countable set is of dimension zero. Note that the dimension of an analytic space can be infinite 6 ) , while each of its points has a neighbourhood of finite dimension. If X is a subspace of an analytic space Y, then the quantities dima E, dim E defined above remain unchanged if E is regarded as a subset of Y. If X is a manifold, the above definitions of dima E and dimE coincide with those given in II. 1. If Z c X is a non-empty submanifold, then dim Z, as defined above, is also the dimension of the manifold Z. Moreover, we have the formula

e

dim E

= sup{ dim r: r c E

is a submanifold of the space X} .

We say that the set E is of constant dimension k (where kEN) if dimz E = k for z E E. The dimension of the germ A of a set at a point a is well-defined by: dim A = dima ii, where ii is a representative of the germ A. All the properties of the dimension listed in II. 1.3-6 7 ) remain true in analytic spaces. They either follow directly from the definition or can be verified exactly as in the case of subsets of manifolds.

e

5. A regular point (of dimension k) of an analytic space X is a point of X a neighbourhood of which is a manifold (of dimension k). A point of X

that is not regular is called a singular point. By X*, XO, and X(k) we denote respectively: the set of singular points of X, the set of regular points of X,

e e

6

en 7

)

3 )

It is enough to take the disjoint union of the spaces z - - t Z E en, n = 1,2, ... ).

With regard to analytic germs, see nO 9 below.

e 1 , e 2 , e 3 , ...

(with the atlas

§4. Analytic spaces

279

and the set of regular points of dimension k of X. In particular, for each locally analytic subset V of an analytic space, we have thus defined the regular points (of dimension k), the singular points, and the sets V*, VO, VCk). In the case when X is a manifold, these definitions coincide with those given in IV. 2.1. For any chart, we have ., are of constant dimension r (because 7r- 1 (z) = z x V z ), so are the sets V;. Hence V;' = D or V;' = 0, depending whether V; =I 0 or V; = 0 (see II. 3.6). Therefore it follows that V" = 7r(V') X D. Thus (see II. 3.4) 7r(V') is an analytic subset of i'>. of dimension dim Va - r (since dim V" = dim Va). THEOREM 1 (REMMERT'S RANK THEOREM ). If f: morphic mapping of analytic spaces such that

X

--+

Y is a holo-

dim lzf = r for z EX, then each point a E X has an arbitrarily small open neighbourhood whose image is locally analytic (in Y) of dimension dimX a - r.

Indeed, one may assume that X, Yare locally analytic subsets of vector spaces NI, N, respectively. Then one can apply lemma 2 to the locally analytic set f C NI x N (see 3.1) and to the natural projection 7r: f --+ N, using the fact that l( ) = (lzj) x f(z) for z EX. This gives us the required result, z,f(z)

because (see 3.4) the natural projection 7ro :

f

--+

X is biholomorphic (and

f(U) = 7rh)l(U)) for U eX). For any holomorphic mapping f: X --+ Y of analytic spaces and for any point z EX, the number p z f = dim X, - dim I z f is called the Remmert rank of the mapping f at the point z. Therefore the dimension of the image (of any arbitrarily small neigh bourhood of the point a) in theorem 1 is equal to the Remmert rank Pal. In the case when X and Y are manifolds, the Remmert rank may be different from the ordinary rank. For inst.ance, if g: C:') z --+ z2 E C, then rankog = 0, whereas Pog with constant rank r, then pzf = r in X (see C. 4.1, t.he the general case, Pz! = rank,! in an open dense subset. see also, 4.8 and II. 3.7). If X and Yare manifolds, then Pz!

=

2:

= 1. However, if ! is a mapping rank theorem). Consequently, in of XO (see 4.5 in ref. to IV. 2.1; we have

rankz! .

=

=

Indeed, consider a E X. By putting k Pal and n dimX, we have dimra n - k, where r = !-I(I(a)). Accordingly (see IV. 2.5), in any neighbourhood of a there is some z E r(n-k). Since! is constant on r, we have Tzr C ker d z ! (see C. 3.11), and so rank,! = codim(ker d,f) :s; codim Tzr = k. Hence, using lower semicontinuity of the function z --+ rankz! (see C. 3.12), we conclude that rank a ! :s; k.

Assume now that the space X is of constant dimension. Then, by theorem 3 from 3.2, the function X :') z --+ Pz! is lower semicontinuous and - by theorem 5 from 3.3-

296

V. Holomorphic mappings

V.6

the set {z EX: pz ::; k} is analytic for each kEN (58). In this case, Remmert's rank theorem can be reformulated as follows.

If the Remmert rank of the mapping f is constant and equal to k (59), then each point of the space X has an arbitrarily small open neighbourhood whose image is locally analytic (in Y) of constant dimension k. We will also show that in the general case (without the assumption that the space X is of constant dimension), rank f = max{pzf; z E X} . Indeed, the equality holds when X is irreducible, due to lower semicontinuity of the functions X :3 z ---> pzf and XO :3 z ---> rankzf. Owing to (*), it is enough to show that if a E X, then Paf ::; rank f. Now, by taking the decomposition X UXi into simple components, we have dimXa dim(Xs)a for some s (see II. 1.6), and so

=

=

Paf

= dimXa -

dimlaf ::; dim(X,)a - dimlafx.

= Pafx.

::; rank fx, ::; rank f ,

(see 4.8 in ref. to corollary 1 from theorem 4 in 3.2).

3. Let .6., n be open convex neighbourhoods of zero in the vector spaces M, N, respectively, and let V be an analytic set in .6. X n. If the natural projection rr: V - - 7 .6. is open and 0 X n c V, then V = .6. X n. LEMMA

PROOF (60). One may assume that dim M = 1. Indeed, suppose the validity of the lemma in this case; then, taking any line L c M, since 0 x n c VL and the projection rrvL : VL - - 7 .6. n L is open (see B. 2.1), we have VL = (.6. n L) x n. Now, suppose that V

o.

Suppose that the set E satisfies the condition (r) near Loo. Let

a E Loo. Then the domain L \ H * of some chart f3 H of the manifold L contains a (see nO 5). Thus, for some neighbourhood U of a, we have the inequality Izl :::; ](gf3H(u,L oo \ H*)-S in the set Eu for some ](,5 > O. Therefore, in view of (***), there is a C > 0 such that

Izl:::; Glul s

for (u,z) E Eu.

(21) For, in the case when (a, b) ~ En(M X Xoo), one must have (a, b) ~ E. If one chooses a sufficiently small neighbourhood ~ x U, it is disjoint from E.

364

VII.4.1

VII. Analyticity and algebraicity

Now a finite number of neighbourhoods U cover L(X) and one can find common C, s for such neighbourhoods. They cover some set {lui 2 R} (see n° 4), and so we obtain the inclusion (###). Conversely, suppose that the inclusion (###) holds. Let a E L(X). Then a = >'(X), where>. E peL) (see nO 3). Consider an affine hyperplane H :j 0 of the space L such that H * -.jJ >.. Then, according to (*), the domain L \ H * of the chart j3 H contains the point a, and I>'H(u)12 2c:lul on>. for some c: > O. The set n = {C(t,u): I>'H(U) > c:lul} c L is open (see B. 6.1), it contains a, and the inequality I>'H(U)I > c:lul holds in n n L. The set D.. = ({luI2 R} U LCX») n n \ H* is a neighbourhood of the point a (see n° 4), and so, by (###) and (***), for (u,z) E E1:J. we have the inequality Izl ::; Clul'::; Cc:-'I>'H(U)I' = CC:-'{!f3H(u,L(X) \ H*)-'. Thus, in conclusion, the set E satisfies the condition (r) near LCX).

§4. Grassmann manifolds Let X be an n-dimensional vector space. 1. Consider the Grassmann space Gk(X), where 0 ::; k ::; n. The homeomorphisms introduced in B. 6.8 cpuv:

L(U, V) :1 f

---4

j

E

n(V),

where

U E Gk(X) and V E n(U)

constitute an inverse atlas on Gk(X) which defines the natural .'Jtructure of a ken - k)-dimen.'Jional complex manifold on the Gra.'J.'Jmann .'Jpace Gk(X), In fact, in order to show that any composition (cpu',v' )-1 0 cpuv is holomorphic, it is enough to check that its graph is analytic in L(U, V) x L(U', V') (see V. 1, corollary 3 of theorem 2; and II. 3.4). Now, the graph is the set

where

r : U:1 u

r

---4

u

+ feu)

E X and ge: U':1 u

(r

+ g(u) E X. This j + g. Therefore the

---4

u

is so because im = j, im ge = g, and im ill ge) = graph is analytic, being equal to the inverse image of an analytic subset under the affine mapping L(U, V) X L(U', V') :1 (f, g) ---4 ill ge E L(U X U ' , X) (see II. 3.2).

r

(22) For F E L(U,X), G E L(U',X), the mapping F$G E L(U x U',X) is defined by (F$G)(u,u') F(u)+G(u'). We have the isomorphism L(U,X) x L(U',X) E (F,G)---> F$G E L(U X U',X).

=

365

§4. Grassmann manifolds

This can also be checked directly, without using the analytic graph theorem. In fact, the mapping : B(U') X (yl)k :3 (tt, v) ----+ f"v E L(U', VI) is holomorphic by the lemma from nO 2 below. Take a basis a1, ... , ak of U, and let p: X ----+ U ' , q: X ----+ yl be the projections corresponding to the direct sum X = U ' + V'. We have the affine mappings

+ f(ad), ... ,p(ak + f(ak))

P: L(U, Y):3 f

----+

(p(a1

Q: L(U,Y) 3 f

----+

(q(a1 +f(ad),···,q(ak +f(ad) E (yll·

Now, ('Pu' v' ) -1

0

'PUV = 0 (P, Q). Indeed, the domains of both sides coincide

if f E p-1 (B(U ' )) and g means that ai

E (U')k,

+ f( ai)

e3 ), and

= (PU), QU)), then g (p(ai + f(ai))) = q(ai + f(ai )),

E g. Therefore g =

j,

which

and thus g = ('Pu' v' ) -1 ('Puv U)).

The natural structure of the manifold G 1 (X) coincide with that defined earlier (see 2.1) for the projective space P(X) = G1(X). Indeed, for any affine hyperplane H ;i 0 and any line .A E D(H.), we have the affine isomorphism r)..H: L(.A,H.) '3 f ~ c + f(c) E H, where c = .A n H. Then 'P)"H. = O'H

0

r)..H.

2. We have the following LEMMA.

Lei Y be a vector space. The mapping Bn(X) X yn '3 (x, y)

fry E L(X,Y), where fry is defined by fxy(Xi)

= Vi,

x

= (XI, ... ,X n ),

~

Y

=

(YI,'" ,y,,), is a (holomorphic) sv.bmersion. Indeed, it follows from the proof of the lemma in D. 6.3 that this mapping is holomorphic. (Since the mapping B1I(X) '3 x ~ F x- l E L(yn, L(X, Y)) is holomorphic; see C. 1.12.) The differential of the mapping is surjective at each point (x,y) E Bn(X) x Y", because the linear mapping yn '3 w ~ fxw E L(X, Y) is surjective. The mapping defined in B. 6.1

is a surjective submersion. Indeed, for any chart 'Pu,v, take the biholomorphic mapping

(23) Since j E O(Y') ~ PU) E B(U'), because both conditions mean that Pj: U' is an isomorphism.

j----+

366

VII.4.3

VII. Analyticity and algebraicity

The composition

is a submersion, by the lemma. Therefore the restriction submersion. Since the sets a is a submersion.

a-I

(n(V))

aa- (o(V») 1

is also a

cover Bk(X) (see B. 6.8), the mapping

As a consequence, we obtain the following characterizations of submanifolds and (locally) analytic subsets of the space G k , as well as holomorphic mappings of those objects into manifolds (see C. 4.2, and also II. 3.4 and V. 3.1 ): A set reG k (X) is a su bmanifold or a (locally) analytic subset (of constant dimension) if and only if the set a-I(r) C Bk(X) is a submanifold or a (locally) analytic subset (of constant dimension), respectively. Then, if M is a manifold, a mapping f: r ~ M is holomorphic if and only if the mapping f 0 a: a-I(r) ~ M is holomorphic. 1fT C X is a subspace of dimension 2 k, then the space Gk(T) C Gk(X) is a submanifold and the induced manifold structure on Gk(T) coincides with the natural one. Indeed, the set Bk(T) = a-I (Gk(T)) C Bk(X) is a submanifold and the mapping K: Gk(T) '-+ Gk(X) is holomorphic, since K 0 aT = (aX)Bk(T)' (See the remark in C. 3.10.) Any isomorphism of Grassmann spaces is biholomorphic: if Y is a vector space and 'P: X ~ Y is an isomorphism, then rp = 'P(k) : Gk(X) ~ Gk(Y) is a biholomorphic mapping. (It is enough to observe that the composition

}), where {Pc>} is a system of Plucker coordinates of the subspace L E rand g: r# --+ M is a holomorphic mapping (satisfying the homogeneity condition g( tp) = p(p) for t E C \ 0, P E r#). By Plucker dual coordinates of a k-dimensional subspace L C c n one means Plucker coordinates of the subspace Ll. of (C n )- identified with c n via the isomorphism c n 3 c ---> (z ---> I:CiZi) E (C n )-. Thus: A system of Plucker dual coordinates of a k-dimensional subspace L C c n is exactly the system of the maximal minors of the coefficient matrix of any system of linear equations of the form Cll Zl + ... + Cln Zn = 0, C

n -k,l Zl

+ ... + Cn-k,n Zz

°

=

that describes this subspace. (Indeed, the left hand sides of the equations are linear forms that constitute a basis for the subspace Ll..) In the same way one can characterize submanifolds of G d C n ) and holomorphic mappings from such submanifolds into manifolds using Plucker dual coordinates. (It suffices to use the fact that T: L ---> Ll. is biholomorphic; see nO 2.) 5. In the space GA, (X) one introduces the structure of a (k

+ 1)( n

- k )-dimensional

manifold by transferring it from Gk+dC x X) \ Gk+1(0 x X) through the bijection X defined in B. 6.11. Then the mapping

---> Z

+

2..: CXi

C(l,z)

+

L

k

f3: X

X

Bk(X) 3 (z, xl,.·., Xk)

E GA,(X)

is a surjective submersion. Indeed, the mapping

X-

1 ofJ: X x Bk(X) 3 (Z,X1, ... ,Xk)

k

--->

C(O,Xi) E Gk+1(C x X)

is holomorphic (see nO 2) and so is the mapping fJ. Let a = (c, a1,' .. , ak) E X Take a linear complement V of the subspace U = 'Y: V k+ 1 3 (w, V1 , ... , v k)

--->

(c + w, a 1

X

Bk(X).

I:~ Cai and the mapping

+ V1 , ... , a k + v k)

EX

X

B k (X) .

371

§5. Blowings-up

Then the holomorphic mapping X-I of3o-y: V k +l --+ 0(0 X V) is bijective (27) and hence biholomorphic (see V. 1, corollary 2 of theorem 2). Therefore the differential dO(X- l of3 o-y) is surjective and so is the differential d a(X- l 0 (3). Thus X-I 0 f3 is a submersion and so is

f3.

The mapping v: G~(X) 3 L --+ L. E Gk(X) defined in B. 6.11 is a surjective submersion, since v 0 f3 a 0 rr and the natural projection rr: X x Bk(X) --+ Bk(X) is obviously a submersion (see C. 4.2).

=

The mapping a: X submersion, because a 0 (e

X X

Gk(X) 3 (x, L) --+ x + L E G~(X) is also a surjective a) = f3, where e is the identity mapping of X (see C. 4.2).

The bijections

where Ll(U, V) denotes the vector space of the affine mappings from U to V, and j = {u + j(u): u E U}, form an inverse atlas on the manifold G~(X). Indeed, the sets v-l(O(V») cover G~(X) and each of the mappings 1/Juv is biholomorphic. This is so because (see V. 1, corollary 2 of theorem 2) it is holomorphic, since 1/Juv(f) f3(J(0), el + j(q) - j(O), ... ,ek + j(ek) - j(O»), where el, ... ,ek is a basis of U.

=

Note also that if U E Gk(X) and V E O(U), then V 3 z ImmerSIOn.

--+

z

+U

E G~(X) is an

Let P = P(X). In the space Gk(P) (0 :::: k:::: n - 1), one introduces the structure of a manifold of dimension (k + 1)(dimP - k), transferred from Gk+l(X) via the bijection w = w P defined in B. 6.12. Then the mapping J1.: G~(X) \ Gk(X) 3 T is a surjective submersion. For J1. (see C. 4.2).

0

f3 Bk+l(X)

=w

0

.-+

(CTr E Gk(P) defined in B. 6.12 = G~ (X) \ G dX)

a and f3(B k +1 (X»)

Finally, observe that the bijection {): Gk(X) \ Gk(X co ) 3 L (see 3.3) is a biholomorphic mapping because {) = X 0 (w X )-1.

--+

L n X E G~(X)

§5. Blowings-up Let X be an analytic space. 1. Let h, ... the mapping

,!k a

be a sequence of holomorphic functions on X. Consider 0

J: X \ 5:1 z --; CJ(z) E

Pk-l ,

(27) Because the sum C X X = (C xU) + (0 X V) is direct and (1, e'l, (0, al)"'" (0, ak) is a basis of the subspace C xU, where e ' E U is the image of e under the projection parallel to V.

372

VII.5.2

where a = a c of its graph

VII. Analyticity and algebraicity k

(see 2.1),

f = (II, ... , /k), and S = V(II,· .. , /k). The closure Y

=a0f c

X x P k-l

is an analytic set, because a

0

f

= E(f) \ (S x Pk-d ,

where

E(f)

= {(z, Cw):

w =1= 0, Wi/j(Z)

= Wjfi(Z),

i,j

= 1, ... , k} c

X

X Pk-l

is an analytic subset (see V. 4.5 in ref. to theorem 5 from IV. 2.10). The natural projection 7r:

Y--tX

(or the pair Y, 7r) is called the (elementary) blowing-up of the space X by means of the functions II, ... , fk. It is holomorphic, proper, its range is

X \ S = {J =1= O}, and the restriction 7r X \s: a 0 f - - t X \ S is biholomorphic (see V. 4.7 in ref. to V. 3.4). Therefore it is a modification of the space X in the set S, provided that f =.:j. 0, i.e., S is nowhere dense (28). It always has the property (m) (see V. 4.11). The set Y is called the blownup space of X by means of the functions II, ... ,fk or, shortly, the blown-up space. The analytic subset S is called the centre of the blowing-up and its inverse image 7r- 1 (S) C Y is called the exceptional set of the blowing-up. lf G c X is an open subset, then by means of the restrictions (fda.

7r

0

is obviously the blowing-up of G

lf V C X is an analytic subset, then the set

(which is analytic (29)) is called the proper inverse image of the set V under the blowing-up 7r. The restriction 7rw: W - - t V is the blowing-up of V by means of the restrictions (fdv. Clearly, 7r- 1 (V) = W U 7r- 1 (V n S). Any blowing-up by means of a single function f 1= 0 is trivial, i.e., 7r is biholomorphic. (In such a case, Y = X x Po and Po consists of a single point.) (28) It can be always achieved by removing the irreducible components of X on which

f =

o.

(29) See, e.g., V. 4.5 in ref. to the theorem from IV. 2.10.

§s. Blowings-up

373

°

2. Consider the blowing-up of en by means of the functions ZI, . .. ,Zn, that is, the blowing-up oj the space en at the point eO). It consists of the set TIn = E(z) = {(z,ew): w -I- 0, WjZi = WiZj} c en x Pn-l (w = (WI' .. " w n )) and the natural projection 7r = 7r n : TIn ---> en ( 1 ). Now, the set TIn is a closed n-dimensional submanifold, for its image under the chart 1/;. = (idcn) x 'P.-l (see 2.3) is the closed n-dimensional sub-

manifold TI(') = {Zi = Z.Wi, i -I- s} c en x en-I. The exceptional set L = 7r- 1 (0) = X P n - 1 C TIn is an (n - I)-dimensional submanifold and cn the mapping 7r \0 : TIn \ (0 x P n-J) ---> en \ is biholomorphic. At each point of the exceptional set (0,7]), 7] E P n-l, we have

°

°

(Because for some s we have 7] = ew, w. = 1; then 1/;.(0,7]) = (O,Wl,' .. ... ,W.-l,W.+l, ... ,w n ), and so 7] X e n - 1 is the tangent space to TI(') at the point 1/;.(0,7]).) Note that if J is a family of ideals generated by ZI, ... ,Zn in en, then its inverse image 7r~ J is a family of principal ideals. Indeed, consider ( E I; (if 7r( () -I- 0, then (7r* J)( = O(TI n )). Then E P n - 1 , and we have im d(7r = 7]. The germs Zi = (Zi)007r( that generate the ideal (7[* J)( belong to the ideal J(I;(), which is principal. But d(z; = Zi 0 d(7r, and so im d(,i; = Zi(7]). Hence d(Zi -Ifor some i. Therefore (see II. 4.2) we must have (7r* J)( = J(I;d.

( = (0,7]), where 7]

°

Finally, note that if 7r is the blowing-up by means of the functions

iI, ... ,jk E Ox (as in nO 1), then we have the commutative diagram y f

--->

where j

= (iI, ... ,jk),

g

= (J x

e)y, e

= idpk_l·

3. Adopting the notation from n° 1, suppose now that X is an ndimensional manifold and j is a submersion at each point of the set S. Thus S is an (n - k )-dimensional submanifold ( 2 ) . eO) See nO 6 below. Hence it is a modification of the space

e

1

)

We have 0

(32) Therefore

X

11":

Pn Y

- 1

en

C E(z) \ (0 x P n ), and so IIn = E(z).

---t

X is a modification in S.

at the point O.

374

VII.5.4

VII. Analyticity and algebraicity

Then Y = E(f) is an n-dimensional manifold, the exceptional set s : 5 x P k-l ---+ X-I (5) = 5 x P k-l is an (n-1 )-dimensional submanifold, x 5 is the natural projection, and xX\s: Y \ (5 x P k-l) ---+ X \ 5 is biholomorphic. Indeed, one may assume that X is connected and I is a submersion 3 ). It is enough to check that E(f) is a connected manifold, because 5 x P k - 1 E(f) (see nO 1). Now, E(f) is the inverse image of the submani-

e

*

fold Ilk C C k

X

P k-

1

under the submersion X

=I

x (idpk_t), and so it is a

sub manifold (see C. 4.2). It is connected, since E(f) = a 0 I U (5 x Pk-I), the set a 0 I is connected, and it intersects z X P k - 1 for each z E 5. So Y = X-I (Ilk), hence we have (see nO 2 and C. 4.2)

and then im d(z,'1)x = (d z J)-I(ry). Therefore, if z E 5, then in VIew of T z 5 = ker dzi we have the biholomorphic bijection 4 ):

e

4. Let us adopt again the notation from nO 1. Let gl, . .. , gl be a sequence of holomorphic functions on X.

e

S ) , then the PROPOSITION 1. If Ii and gi generate the same ideal blowing-up by means of Ii is isomorphic to the blowing-up by means of 9j.

PROOF.

f3 0 9 with the nat(see 2.1) and 9 = (gl, ... ,91).

The blowing-up by means of 9j is the set Z =

ural projection 7r': Z

---+

X, where

We have 9j = L:7=1 aijIi and Ii in X. Thus g(z)

=

f3 =

aCl

= L:~=1 bijgj, where aij, bij

a(z,I(z)) and I(z)

e e

=

are holomorphic

b(z,g(z)) ,

3 ) It suffices to prove the above properties for the connected components of a suitable neighbourhood of the set s.

4 ) If x: L ----+ C k is a linear mapping (of a r-dimensional, then the mapping Pk 3 >.. ----+ X- l Indeed, if L = N + M is a direct sum, then this biholomorphic mapping (X;/ P k ----+ P(M) (see P(M) E J1. ----+ N + J1. E sr+l (N, L) (see 4.3).

r:

e

S

)

In the ring

Ox,

i.e.,

L:Ox/;

= L:0Xgj.

vector space L) whose kernel N is

(>..) E sr+I(N, L) is biholomorphic. mapping is the composition of the 4.2) and the biholomorphic bijection

375

§s. Blowings-up where a(z, u)

= (L:~ ali(z)ui, ... , L:~ a1i(z)ui)

and b(z, v)

= (L:~ b1jvj, ...

...,L:~ bkj(z )Vj). Consider the holomorphic mappings defined by a: G:3 (z, Cu)

-+

(z, Ca(z, u)) E X

b: H:3 (z,Cv)

-+

(z,Cb(z,v)) E X x P k -

= ((z,Cu): a(z,u) =I O} C X c X X Pl- 1 (36). In view of (*), 0 f, and hence also when (z, Cu)

where G

b(z,v) =I O} (z, Cu) Eo:

are

0

g -+

ay: Y

-+

0: 0

f

1 ,

E

Y. This implies that Y C G.

ao: o!:

0: 0

f

-+

(30 g and

are mutually inverse biholomorphic mappings, and so

Z and

bz: Z Y

commutes. Therefore

Pl- 1

X Pk-l and H = {(z,Cv) : we have b(z,a(z,u)) = u when

Similarly, Z C H. Now using (*) we conclude that bf3og: ~

X

7r

-+

Y. Clearly, the diagram

ay

-+

Z

~ 7r'.

COROLLARY. If fi and gj are generators of the same family of ideals Iz C Oz, z E X (see VI. 1.2), then the blowing-up by means of fi is isomorphic to the blowing-up by means of gj.

In fact, let 7r: Y - + X, 7r': Z - + X be the blowings-up by means of fi and gj, respectively. Each point z E X has an open neighbourhood U z

such that (ji)U z and (gj)u z generate the same ideal, and so 7r uz ~ (7r')U z . Since 7r and 7r' have the property (m) (see nO 1) which is sublocal and rigid, we have 7r ~ 7r' (see V. 4.11-12). 5. A blowing-up of the space X by means of the family of ideals Iz C z EX, is defined to be a holomorphic mapping 7r: Y - + X of an analytic space Y (or, also, as a pair Y, 7r) such that each point of the space X has an open neighbourhood U for which tr u is isomorphic with the blowingup by means of generators of the family I z , z E U. Therefore the family I z , z EX, must be coherent. The space Y is called the blown-1lp space of X by means of the family {Iz} or, shortly, the blown-up space.

o z,

In particular, the blowing-up by means of the holomorphic functions a blowing-up by means of the family of ideals generated by

it, ... ,fk is it,···,fk.

e

) Since e x a, e x f3, where e = id x , are surjective su bmersions (see 2.2), the sets G, H are open and the mappings ii, bare holomorphic (see C. 4.2). 6

376

VII.5.5

VII. Analyticity and algebraicity

All blowings-up are proper mappings

e

7

).

All blowings-up have the property (m) which is sublocal and rigid (see nO 1). Thus, according to the corollary of proposition 1 in nO 4, it follows (see V. 4.12) that: All blowings-up by means of the same family of ideals are isomorphic. In particular, every blowing-up by means of the family of ideals generated by II, ... , fk is isomorphic to the blowing-up by means of II, ... , fk.

If 7r : Y ~ X is a mapping between analytic spaces and X = an open cover, then obviously:

UG

t

is

is the blowing-up by means of the family {Iz}) -¢:::::} Gt -¢:::::} (each 7r is the blowing-up by means of the family {Iz} zEG, .)

(7r

PROPOSITION 2. For any coherent family of ideals Iz C Oz, z E X, there exists a unique (up to an isomorphism) blowing-up of the space X by means of this family.

Indeed, there is an open cover X = U Ut such that for each L the family {IZ}ZEU, has generators It, ... ,ft. Let 7r t : Y, ~ Ut be the blowingup by means of the functions It, ... , ft. For any pair L, 11:, the restrictions

7ry,nuKand 7r~,nuK are the blowings-up by means of U,')u,nu~ and Unu,nu.,

respectively. Consequently, in view of the corollary of proposition 1 in nO 4, it follows that 7ry,nu. ~ 7r~,nu.:

UX

(z, >.)

X. It follows that the X -conic set E

V[r}

C

UX X

is analytic in U X X (50). In addition, it is of constant dimension n, since (see IV. 2.5) the set 1/>-1 (V[r) = S \ (U X 0), which is open and dense in S, is of constant dimension n (see II. 3.4). We have (a,O) E S. Thus the germ S(a,O) is of constant dimension n, and so S(a,O) = V(j), where f E O(a,O) \ is without multiple factors (see II. 5.3). Let F be a holomorphic

°

representative of f in an open neighbourhood U' X 6' C U X X of the point a such that S n (U' X 6') = V(F) and F = I:: F" is uniformly convergent in U' X 6'. Here the F" are holomorphic in U' X X and X -homogeneous of degree v, v = 0,1, ... (see C. 3.18). One must have SUt C V(F,,) (see footnote (47a)), and hence (see the theorem in II. 5.2) each of the germs (F,,)(a,O) is divisible by f. Therefore, by the corollary of the proposition from II. 3.8, there is a neighbourhood n C u' X 6' of the point (a, 0) such that (F,,)n = G"Fn , where G" are holomorphic functions on n (v = 0,1, ... ). Then I: G" = 1 and the series is almost uniformly convergent in n \ S. Hence, the series is also uniformly convergent in n (see II. 3.9). It follows that Gk(a,O) i- 0 for some k. Taking an open neighbourhood U X 6 C n of (a,O) in which Gk i- 0, we have S n (U X 6) = V((Fk)UX~). This implies

n:

that Su = V((Fk)UXX) (since both sets are X-conic). Thus (in view of the equivalence (z, A) E Vu .), >. E P(M) (see 4.3), we have dim~ = dimp(7r- 1 (Voo )) ::; dim 7r- 1 (Voo) ::; ken - k -1) + dim Voo (see II. 1.4). Therefore dim~ < ken - k) = dimGn-k(M), which proves (see II. l.2) that ~ is nowhere dense. On the other hand, if dim V > k, then dim V00 ::::: k, and for each Y E Gn-k( M) we have dim Y00 + dim V00 ::::: n - l. Thus (sec 6.2) \ve must have Yoo n Voo #- 0, which means that Y is not a Sadullaev subspace for V. COROLLARY. For every algebraic subset V C M, there exists a projection 7r: 111 ---+ X onto a subspace X C M whose restriction 7rv: V ---+ X is finite and surjective. If, in addition, V is of constant dimension, then 7rv is an open *-covering (see V. 7.2). 2. Let X and Y be vector spaces. PROPOSITION l. Let V C X x Y be an algebraic subset. If the natural projection V --> X is proper, then (after selecting norms on X and Y), we have V c {(x, y): Iyl::; M(I + Ixkl)} for some M, k > 0 . Indeed, the closure if of the set V in X x Y is algebraic (see 6.4), and hence the pair if, X x Y00 satisfies the condition of regular separation (see the theorem in IV. 7.1). Therefore, by lemma 1 from 3.6, the set V satisfies (51) It is even the complement of a nowhere dense algebraic set. See the proof, and 17.1314 below (e.g., Chow's theorem). (52) See E. Fortuna [22a].

390

VII.7.3

VII. Analyticity and algebraicity

the condition (r) near X oo , and so, by lemma 2 from 3.6, we get the required inclusion. According to Liouville's theorem (see C. 1.8), we have the following corollary (a special case of Serre's theorem; see 16.3 below): COROLLARY 1. Every holomorphic mapping X is a polynomial.

---4

Y with algebraic graph

COROLLARY 2. (53). The inverse of any biholomorphic polynomial mapping is a polynomial. 3. Let M be a manifold, and let Z C Ai be a nowhere dense analytic subset. Let N be a vector space.

PROPOSITION 2. Let V be an analytic subset of (lv! \ Z) x N of constant dimension m = dim M such that the nai1Lral projection V ---4 M \ Z is proper. Let V be the closure of V in AI[ x N. Then the following conditions are equivalent:

(1) the set V satisfies the condition (r) near Z;

(2) the pair V, M x Noo satisfies the condition of regular separation; (3) the set V is analytic in M x N. PROOF (54). We already have the implications (3) ==? (2) ==? (1) (see the theorem in IV. 7.1, and lemma 1 in 3.6). It remains to show the implication (1) ==? (3). Let a E M, and assume the inclusion (r) from 3.6 with E = V and X = N. It is enough to show that for some open neighbourhood U of a the closure of Vu in U x N is analytic (see II. 3.4). Clearly, one may assume that M is an open subset of a normed vector space, and that e

1, and assume that the statement is true for n - 1. We have

E = (En(B x C») uU~ En{fiO = O}, where B = {z: flO

# O, ... ,flO # O}.

By the

induction hypothesis, the sets r.(E n {fiO = O}) are constructible. Furthermore, lemma 2 and the elimination lemma imply that the set

r.(En(B

X

C»)

= {z

E B: (Jll/iro, ... ,irn,/iro, ... ,flIlflO, ... ,Jln,lflO)(Z) E r.(0)}

is also constructible. This completes the proof.

398

VII.8.4

VII. Analyticity and algebraicity

Let X and Y be vector spaces.

If Fe XxY is a constructible set, then the sets {z E Xi #Fz ;::: k}, k = 1,2, ... , are constructible. In particular (64), if f E - - t Y, where E C X, is a mapping with constructible graph (65), then the sets {W E Y: #f-1(w);::: k}, k = 1,2, ... , are constructible. LEMMA.

Indeed, the k-th of these sets is the image by the projection X x Y of the constructible set

{(z, WI, ...

, Wk)

E X x Y: (z, wJ), ... , (z, wk) E F,

WI, ... , Wk

--t

X

are distinct}.

4. In what follows, let X be an n-dimensional vector space.

In view of proposition 2 from n° 3, we have analogues of proposition 7 and corollary 2 from IV. 8.4 for constructible sets (in X). A set E C X is constructible if and only if the sets V;(E) are algebraic and VB = 0 for some s. Then Vi+1(E) is nowhere dense in V;(E), i = 0,1, ... , we have ViCE) = 0 for i > n, and if 2r > n, then

E = (Vo(E) \ Vl(E)) U ... U (V2r - 2 (E) \ V2r - 1(E)) The constructible sets are precisely the sets of the form (Vo \ Vr) U ... U (V2k \ V2k+l), where Vo ~ ... ~ V2 k+l are algebraic subsets such that V;+1 is nowhere dense in Vo (i = 0, ... ,2k). In other words, every constructible set E C X has a decomposition into (disjoint) quasi-algebraic subsets S;. Namely

(*)

E

= So U ... U Sk,

where S;+l is nowhere dense in

as;, i = 1, ... ,k

.

°

The decomposition (*) is unique (assuming that Sk =I- 0 if E =I- 0, and k = if E = 0). This is so because for a decomposition (*) one must have So = E\

(E\E) (66). 5. By a constructible leaf we mean a non-empty connected submanifold ar are

reX which is a constructible set or, equivalently, such that f' and (64) By interchanging X and Y.

(65) Then E, being the image of the graph of constructible.

f

by the projection X X Y

---+

X, is also

(66) Indeed, E = So :> ... :> Sk, so 51 U ... U 5k is nowhere dense in 850. Hence

E\ E

= 850 \

(51 U ... u 5k) is dense in 850, which means that 850

= E \ E.

399

§8. Constructible sets

algebraic. (Thus a constructible leaf is always quasi-algebraic.) By proposition 1 in n° 3, every analytically constructible leaf in X is a constructible leaf (see IV. 8.3, corollary 1 of proposition 5). By a constructible stratification of an algebraic set V we mean a finite partition of V into constructible leaves r~ such that dim r~ = i and each set 8r~ is the union of some r~, i < k (67). For any finite family of constructible subsets of the space X, there is a constructible stratification of X that is compatible with this family. Indeed, it is enough to take the complex stratification {r~} of the projective closure X which is compatible with this family and with the set Xoo (see IV. 8.4, proposition 6). Then the leaves r~ c X form the desired stratification. In particular, every algebraic set has a constructible stratification.

The connected components of a constructible set are constructible sets; there is a finite number of them. Finally (see n° 3 proposition 1, and IV. 8.5), if a set E C X is constructible, then so are the sets EO,E*, and E(k) (k = 0, ... ,n) (68). The connected components of the set EO are constructible leaves. 6. Let X and Y be vector spaces. PROPOSITION 3. If f: E - ; Y, where E C X, is a mapping with constructible graph (69), then there is a smooth q1Lasi-algebra.ic set H C E which is open and dense in E and such that the restriction fH: H - ; Y is holomorphic.

In fact, it is sufficient to apply the proposition from V. 5.2 to the manifolds X, Y and the mapping f E - ; Y (see n° 3, proposition 1 and corollary 1 of proposition 2).

7. We will also prove that the graph of the mapping

a = af : Bk(X)->

Gk(X) (see B. 6.1) is analytically constructible in both (X)k x Gk(X) and -k

X

x Gk(X). First observe that in the case k = 1, the graph ofthe mapping a: X\O 3

(67) If the stratification is compatible with an algebraic set W C V, then the leaves r~ C W form a constructible stratification of W.

(68) Since E(k) C (E)Ck), the regular points (of dimension k) of a constructible set can be characterized in terms of polynomials. See 1.7. (69) Then, according to the Chevalley theorem, the set E is also constructible.

400

VII.9.1

VII. Analyticity and algebraicity

Cz E P(X) is constructible in X x P(X). In fact, a = {F(), x J.1-) = O} \ {G()' x J.1-) = O}, where F(t,z,w) = z - tw and G(t,z,w) = tz. z

--+

a = af x

Xk:3 (ZI, ... ,Zk) --+ ZI/\ ... /\ Zk E AkX and p: Gk(X) --+ Gk(X) is the Plucker biholomorphic mapping. Thus (see n° 1) the graph of the mapIn the general case, we have p

ping po a, that is, (e

X

0

0

I, where , :

p)(a) (where e is the identity mapping of Xk),

is constructible in Xk x peAk X), and hence (see IV. 8.3) it is analytically constructible in Xk x Gk(X). Therefore the graph of a is analytically constructible in Xk x Gk(X) (see IV. 8.3). Similarly, one verifies that the graph of a is analytically constructible in (X)k X Gk(X),

§9. Ruckert's lemma for algebraic sets Set P n = p(cn). After the appropriate identifications we have

c = Po

C PI C ... C P n

(see I. 1.1). By Zi we will also denote the polynomial (z 1, ... , n. Thus P n = C[ZI"'" znl.

--+ Zi) E

Pn ,

Z

=

1. Fix an ideal I of the ring P n .

j E P n / I. Denote by PI the image of the subring PI, and set P = 'LJLpTp E Pn[T1 , ... Trl for P = We have the natural epimorphism P n :3

f

--+

1

L: apTP E Pn[T1 , .. . , Trl· (We have the natural epimorphism P n [T1 , . .. , Tr] P --+ P E Pn[Tl, ... ,Trl.) Clearly, P(gl, ... ,grr = PU!l, ... ,[/r) for gi

:3

E

P n , and, more generally, P(Ql, ... ,Qrr = P((h, ... ,Qr) for Qi E P n [SI,,, ... ,Sq]. Let 0 ,:S: k ,:S: n. The ideal I is said to be k-regular if it satisfies the following conditions

(1) I contains a polynomial from PI which is monic in

Zl,

l = k + 1, ... , n;

(2) In Pk = O. This definition implies (see A. 3.3) that every proper ideal is k-regular for some k, after a suitable linear change of coordinates. The condition (1) is equivalent to each of the following conditions: (1') Zk+l, ... in are integral over

Pk,

401

§9. Ruckert's lemma for algebraic sets

(I") P n is finite (and hence integral

eO») over Pk.

Indeed, the condition (1) means that the element ZI is integral over

PI-l (l = k + 1, ... , n). Hence (I") = } (1) and also (1) = } (1'), because PI = PI-dzd (applied repeatedly) implies that the elements Zk+l, ... , zn are integral over Pk (see A. 8.1). Finally, since P n = P[Zk+l, ... ,Zn], we obtain the implication (I') = } (I") (see A. 8.1). Note also that the condition (2) is equivalent to

(2') Pk :3 f

--+

j

E P k is an isomorphism.

If the ideal is k-regular, then it has a finite system of generators from

PdZk+l, ... , ZnJ. (One shows this as in III. 2.2.) 2. Any linear change of the coordinates U = (Zl,'" , Zk) and any linear change of the coordinates v = (Zk+l, ... , zn) does not change k-regularity of the ideal I. One can verify this statement as in III. 2.6, using the conditions (I") and (2). 3. Suppose now that the ideal I is prime and k-regular.

Then (see A. 8.2) the element Zj, j polynomial i)j E PdT] over

=

Pk , where Pj

k+1, ... , n, has a (unique) minimal E PdT]. Obviously,

and so

V(I) C

{Pj(u,Zj)

= 0,

j

= k + 1, ... ,n}

It follows that the projection V(l):3 (u,v)

--+ U

.

E C k is proper.

As was done in III. 3.2, one can choose a linear change of the coordinates

Zk+l, ... ,Zn in such a way that Zk+l is a primitive element of the extension

Pn

of the ring

Pk . We have the following

PROPOSITION.

There is a linear change of the coordinates Zk+l, ... , Zn

which makes Zk+] a primitive element of the extension P n of the ring P k . Moreover, if 0 E Vel), one may require that

(1) CO) See A. 8.1.

v(I)n{u=o, Zk+l=O}=O.

402

VII.9.3

VII. Analyticity and algebraicity

Furthermore, if E C en \ V(I) is a finite set, one may require that V(I)n7r-I(7r(E)) =0,

(2)

where

7r: z-+(z,zk+d.

As for the condition (1), observe that the set V(I) n {u = O} is finite (because of the inclusion (** )). Thus V(I)n{ U = O} \ = {(O, vd, . .. , (0, vs)}. There is a linear form r} C Gn-k(P), r = p - 1, p, each of which is the image by the natural projection of the set {(L,

ZI, ... , Zr+d:

Zi

E L n V are distinct} C Gn-k(P)

X pr+I ,

which is analytically constructible. This is so because the set {( L, z): Z E L} C Gn-k(P) X P is analytic, which follows from the analyticity of S~-k+I(y); see 4.3 and 4.5.) COROLLARY. If V C P is an algebraic set of dimension k (where 0 :S k :S n), then the set {L: L n V = 0} C Gn-k(P) is open and dense with analytic complement (93).

Indeed, one may assume that the set V is of constant dimension l < k (by considering its simple components). Now, the set {L: L n V = 0} is non-empty. This is so, since N n V is finite for some N E Gn-I(P), and hence there is an L E Gn-k(N) such that L n V = L n (N n V) = 0 (94). The complement of our set is analytic because it is the image under projection of the analytic set {(L, Z): Z E L n V} c Gn-k(P) X P. If V c X is an analytic set of constant dimension k, then so is V eX, and deg V = deg V. (Indeed, #(L n If) = #(L n V) for LEG', and the image of G' under the mapping L ---4 L is open and dense in Gn-k(X); see 5 4.5 ).)

e

Let V, W be algebraic sets of constant dimension k in an n-dimensional projective or vector space. The definition of the degree (with the lemma and proposition 9) implies the following properties. If V

of 0,

then deg V > O. (See 6.2 and 7.1.)

If V = Vu ... U Vr is the decomposition into simple components, then deg V = deg VI + ... + deg Vr . (Indeed, take an (n - k )-dimensional subspace L that intersects V transversally. Then it intersects each of the sets V; transversally - see corollary 4 from theorem 4 in VI. 2.9 - and we have #(L n V) = #(L n VI) + ... +

#(L n Vr ).)

Therefore: If V* W, then deg V


O} is closed in W (see V. 3.2, the semicontinuity theorem), and so the set Z 7r(Wa ) is closed. In view of the Bezout theorem, it is the image by the projection onto M of the constructible set

=

{(F,Z1, ... ,Zk)EMxy k ; F(Zi) =0, zitO, ZiAzjtOforitj}, where k = (k 1 , ... , k n )

+ 1.

By the Chevalley theorem (see 8.3), it is constructible and

hence algebraic (see 8.3, proposition 2). Finally, ZC# M because (;', ... , (~n) E M \ Z for any linearly independent forms (1, ... , (n E Y·.

3. Let X be an n-dimensional vector space. LEMMA 3. Consider the mapping F = (FI , . .. ,Fn) X - - t en, where FI , ... ,Fn are polynomials of degrees at most k I , ... , k n 2: 0, respectively.

Let f" denote the homogeneous component of degree k" of the polynomial F" (v = 1, ... , n). If V(h, ... ,In) = 0, then the set F-I (0) is finite and v(F) = kI ... k n . PROOF.

when

Izl

--t

Fix a norm In X. Then F,,(z)/lzlky - f,,(z/lzl) - - t =, and inf {2:~ 1/,,(z)l: Izl = I} > 0. It follows that

°

°

if Izl is sufficiently large. Therefore the set F-I(O) is bounded and thus, being analytic, it is finite: F- I (0) = {aI, ... , aT}' 2:~ 1F,,(z)l/lzlky

>

e17) It is enough to take the set E = H n (F- (0) - V), since it is contained in the union 1

of the compact sets H n (-V), H n (Ai - V) denotes the projection that is parallel to Ai.

= 7ri(-V), i = 1, ... ,no Here 7ri

(118) Obviously, the space M is finite dimensional.

;

Y

--+

H

434

VII.13.4

VII. Analyticity and algebraicity

Let G" E P(C x X) be the form of degree k" such that

F,,(z)

= G,,(l, z)

in X (v

= 1, ... , n)

(see A. 3.2). Consider the system of equations

GI(t,z) = ... = Gn(t,z) = O.

(B')

Every line in P(C x X) is either of the form C(O, a) where a E X \ 0, or C(l, a), where a E X. In the former case, it cannot be a zero of the system (B') (for otherwise one would have a E V(h, ... ,ik)). Therefore - in view of (*) - the only zeros of the system (B') in P( C x X) are the (distinct) lines C(l, ad. Their multiplicities are equal to ma;F (see nO 2), and so the Bezout theorem yields v(F) = I:~ ma;F = ki ... k n . (119). Consider a mapping F =

THE RUSEK-WINIARSKI INEQUALITY

(FI, ... ,Fn ): X ---7 C n , where FI, ... ,Fn E P(X) are polynomials of degrees k l , ... , k n ;::: 0, respectively. If the set F-1(0) is finite, then v(F) ~ ki .,. k n . PROOF. One may assume that kl' ... , k n > O. Let N' be the vector space of all the mappings H = (HI"." Hn): X ---7 en, where HI, .. . ,Hn E P(X) are polynomials of degrees at most k l , ... , k n (120). Let us employ the notation from lemma 1. The mapping cp: N'::1 (HI, ... , H n) ---7 (hI, ... ,h n ) E N, where hi denotes the homogeneous component of degree k i ofthe polynomial Hi (i = 1, ... , n), is an epimorphism. Therefore it is an open and continuous surjection (see B. 5.2). By lemma 1, the set 2:;' = cp-I(2:;) C N' is nowhere dense (see B. 2.2). By lemma 3,

(#)

the set H- I (0) is finite and v(H)

= ki ... k n

for HEN' \ 2:;' .

= {z EX: Izl < s} (after endowing X with a norm). We have F- 1 (0) C Br for some r > O. Put B = B 2r . The set E = fJ \ Br is compact, its interior is non-empty, and so IHI = sup{IH(z)l: z E E} is a norm on the space N'. Since c: = inf{lF(z)l: z E E} > 0, we have IH(z)1 ;::: tc: in E, provided that IH - FI < tc:. Consequently, by taking the open ball Set Bs

D={H: IH-FI 0 (see C. 3.19). This implies that 11/J(z)1 :::; J«1 + Izl) in C for some J< > O. Thus, in view of Liouville's theorem (see C. 1.8), the restriction 1/Jc: C ----t C is linear. LEMMA 5. Any biholomorphic mapping r.p: P(X) ----t P(Y) thai maps every hyperplane onto a hyperplane must be an isomorphism of projective spaces. One may assume that P(X) = M and P(Y) = N, where M,N are n-dimensional vector spaces, and that r.p(Moo) = Moo, r.p(0) = 0 (see 3.1). Then r.p(M) = N. If r.p maps the (k + I)-dimensional (projective) subspaces onto (k + 1)-dimensional subspaces, then the same holds for the subspaces of dimension k (see B. 6.12). Therefore r.p maps lines onto lines. Consequently, if A E P( M), then). C M is a projective line and so is r.p().) eN. Hence PROOF.

= fl, where ji, E peN), and r.p(Aoo) = ji,oo (see 3.3). Thus, by lemma 4, the mapping r.p)": A ----t ji, is linear. Consequently, the mapping r.p M: M ----t N is homogeneous of degree 1, and hence (see C. 1.8) it must be linear. Therefore it is a linear isomorphism, and so (see 3.2) r.p is an isomorphism of projective spaces. r.p().)

437

§14. Meromorphic functions and rational functions

LEMMA 6. If hyperplanes HI"'" Hn C P(X) intersect at a singe point (i.e., #(HI n ... n Hn) = 1), then they intersect transversally. In fact, one may assume that P(X) = iII and HI n .. .nHn = 0, where M is an n-dimensional vector space (see 3.1). Now, HinM C M are hyperplanes (see 3.3) and their intersection at 0 is transversal (see A. 1.18). This implies that the hyperplanes Hi intersect transversally. THEOREM. Every biholomorphic mapping of the space P(X) onto the space P(Y) is an isomorphism (of projective spaces) (124). PROOF. Let r.p: P(X) - - - t P(Y) be a biholomorphic mapping. In view of lemma 5, it is enough to prove that r.p maps hyperplanes onto hyperplanes. Let H C P(X) be a hyperplane. There are hyperplanes H = H J , •• • , Hn such that #(HI n ... n Hn) = 1 (125). By lemma 6, they intersect transversally. Their images r.p(Ht}, ... ,r.p(Hn) C P(Y) are closed submanifolds of dimension n - 1 which intersect at a single point, transversally at that point. In view of Chow's theorem (see 6.1), they are proper principal algebraic sets (see 11.6). Denote by rI, ... ,r n , respectively, their degrees. By the Bezout theorem (from nO 4), we have 1 = rl ... rn, and so rl = 1. Consequently, r.p(H) is a hyperplane (see 11. 7). REMARK. In particular, the biholomorphic mappings of the Riemann sphere C = PI onto itself are precisely the mappings h: P I :3 C(t,z) - - - t C( at + bz, ct + dz), where ad - be i O. In other words (after the identification [: C U CXl - - - t PI, where [(z) = C(l,z), [(CXl) = 0 x C; see 3.1): h(z)

=

c+ dz - - b for z a+ z

i

-bfa, h(-b/a)

= CXl, h(CXl) =

d/b,

in the case when b i 0, and h(z) = c'

+ d'z, h(CXl) = CXl,

where d'

i

0

otherwise. Therefore they are the so-called homographies (see [5], Chapter

IV, §8). This can be also derived directly from lemma 4. Namely, if f: C ---t C is biholomorphic, one may assume that f(O) = 0 and f(=) = = (126). Then the restriction fe: C ---t C is linear, and so f is a homography.

(124) It is a special case of the theorem on biholomorphic mapping of factorial sets (see 18.2 below).

(125) It suffices to take>. E H and hyperplanes H- = L 1 , ... , Ln C X such that n~ Li = >., and set Hi = Li. (126) Due to the fact that the homographies constitute a group, and if ex, {3, -y, DEC, ex ::f:. (3, and -y

:I: D,

then there is a (unique) homography h such that h(ex)

= -y and h({3) = D.

438

VII.l4.1

VII. Analyticity and algebraicity

§14. Meromorphic functions and rational functions Let M be a complex manifold of dimension m > O. 1. We say that a function f is holomorphic nearly everywhere on M if f is a holomorphic function on an open dense subset of the manifold M. The complement Z C M of that set is closed and nowhere dense, and is called the exceptional set for the function f. A point a E Z is said to be a removable singular point if the function f extends holomorphically across this point (see II. 3.8, footnote (12)). Otherwise, it is called a singular point. The set of singular points of f is closed and nowhere dense. We denote it by Sf.

Let O~ denote the set of functions that are holomorphic nearly everywhere on M. By the complete elements of the set 0~1 we mean its maximal elements with respect to inclusion. A function f E O~ is complete exactly when Sf is its exceptional set. We say that the functions f, 9 E O~ are equivalent, and write f ~ g, if they coincide on the intersection of their domains. Obviously, it is an equivalence relation in the set O~. The equivalence class of a function

j with respect to inclusion. f ~ g, then Sf = S9' In fact, j = U{g E 0~1 tional set is Sj = Sf. Let

f E

SIc = Sf

O~.

f E

O~

contains the greatest element

It is the only complete element of the class. If 9 ~

J} is a function from O~ whose excep-

If GeM is an open set, then clearly

fa E 0 0 and

n G. If f is complete, so is fa. If {G,} is an open cover of the

manifold M, then: (J is complete)

~

(all

fa, E 0 0• are complete).

2. If 9 and hare holomorphic functions on M and h ;f 0, then by the meromorphic fraction 9 / h on M we mean the function {h =I- O} :3 z ---t g( z)/ h( z) E C. It is holomorphic nearly everywhere on M and its exceptional set is V( h). If 9 / hand g' / h' are meromorphic fractions on M, then obviously

g/h ~ g'/h' ~ gh' = g'h . LEMMA 1. Let 9 and h be holomorphic functions on a neighbourhood of a point a E M. If the germs ga, ha are relatively prime and ha =I- 0, then for a sufficiently small open neighbourhood U of the point a we have the following properties:

§14. Meromorphic functions and rational functions

(1)

439

1=

gu / hu is a meromorphic lraction on U which is complete: Sf = V(hu ),

°

(2) il c E Sf, then the limit limz---+c I(z) is equal to depending whether g( c) i- or g( c) = 0,

00

or it does not exist,

(3) dim{z E Sf: g(z) = o} < m-l. Indeed, we have dim V(ga, h a ) < m - 1 (see II. 5.3), and hence, if U is a sufficiently small open neighbourhood of a, then the function 1 = gU / hu is a merom orphic fraction on U whose exceptional set is Z = V( hu) and dim(Z n W) < m - 1, where W = V(gu). Since dimz Z 2 m - 1 and dimz W 2 m - 1 for z E Z n W (see II. 5.1), the set Z n W is nowhere dense in both Z and W (see IV. 2.5). Now, if c E Z \ W, then obviously limz---+c I(z) = 00. If c E Z n W, the limit does not exist, since in the latter case the function 1 takes the value zero in any neighbourhood of c (in W \ Z) and it attains arbitrarily large values (sufficiently close to points of the set Z \ W). This implies that Z = Sf. A function 1 E O:W is said to be meromorphic on M (or a meromorphic element 01 the set 0:W) if each point a E M has an open neighbourhood U such that the restriction lu is equivalent to a meromorphic fraction 9 / h on U. (Therefore the equivalence class of any meromorphic element of O:W consists of meromorphic elements only, and it contains the unique complete meromorphic function on M; see nO 1.) Taking a smaller neighbourhood U and replacing the fraction 9 / h by an equivalent one, we can make the germs ga, ha relatively prime and the fraction g/h complete: this is achieved by (*) (in view of the factoriality of Oa) and by lemma 1. Then lu C g/h (see n° 1). Thus we conclude that: A function 1 E 0:W, whose exceptional set is Z, is meromorphic if and only if each point a E Z has an open neighbourhood U such that

fez)

=

g(z)/h(z)

and

h(z)

=I 0

in U \ Z ,

where 9 and hare holomorphic functions on U. In addition, one may require that the germs ga, ha are relatively prime and the meromorphic fraction g / h on U is complete. Next (see also lemma 1 and nO 1), for a function f defined on a subset of the manifold M, the following conditions are equivalent:

(1) 1 is a complete meromorphic function on M; (2) each point a E M has an open neighbourhood U such that lu is a meromorphic fraction of the form g/h on U, where the germs ga, ha are relatively prime;

440

VII.14.3

VII. Analyticity and algebraicity

(3) each point a E M has an open neighbourhood U such that fv complete meromorphic fraction on U. Suppose that f,g E IS

O~

IS

a

and f c::: g. Then if f is meromorphic on M, so

g.

Let f E O~, and let GeM be an open set. Let {G,} be an open cover of the manifold M. If f is (complete) meromorphic on M, then f G is (complete) meromorphic on G. The function f is (complete) meromorphic on M if and only if f G. is (complete) meromorphic on G, for each L. Now we are going to show that for any meromorphic function f on M the set of its singular points Sf is analytic of constant dimension m - 1. If c E Sf, then the limit limz~c fez) is either equal to 00 or it does not exist. In the former case, c is called a pole of the function f, while in the latter 27 ) . The set of indeterminate case, it is called an indeterminate point of f points of f is analytic of dimension < m - 1, and so it is nowhere dense in Sf· (Hence the set of poles of f is open and dense in Sf·) If f c::: g, then f and g have the same poles and the same indeterminate points.

e

Indeed, for any point a E M, there is an open neighbourhood U and a complete meromorphic fraction fa = g / h on U containing fv, and such that the germs ga, ha are relatively prime. Then Sf n U = Sfu = Sfo = V(h) (see nO 1). Therefore the set Sf is analytic of constant dimension m - 1. In view of lemma 1, after taking a smaller U, if c E Sf n U then since the graph of fv is dense in the graph of fa, we have ei ther that limz~c f( z) = limz~c fa (z) = 00 or neither of the two limits exists. Next (see (3) and (2)), the trace on U of the set of indeterminate points of f is analytic (in U) of dimension < m - 1. The last property follows from the fact that the graph of f n g is dense in both f and g. It is easy to check that the family MM of all complete merom orphic functions on M, with addition and multiplication given by

is a ring

e

29 ).

(127) See the corollary from theorem l' in nO 4 below.

+ g and f g are defined in the intersection of the domains of the f, g and are meromorphic on M.

(128) The fractions f functions

(129) For any a EM, we define the field M a of meromorphic germs at a as the field of fractions of the ring Oa. The meromorphic germ at a of a meromorphic function f on M is well-defined by the formula fla = galha, where glh ~ hu is a meromorphic fraction on an open neighbourhood of a (see (*)). One defines the sheaf of meromorphic germs on M as the set S)J1 = U{Ma : a E M} furnished with a suitable topology and the natural projection S)J1

--+

M (see, e.g., [33]'

§14. Meromorphic functions and rational functions

441

If the manifold M is connected, then MM is a field (130).

f E is complete meromorphic if and only if the set Sf is discrete and in a neighbourhood of any of its points a the function f is of the form 3. In the case when the manifold M is one-dimensional, a function

O~

This condition is required to hold in some (and thus in each) coordinate system at a, with a holomorphic h in a neighbourhood of zero such that h(O) i= o and k > 0 (then we have (m) with a holomorphic hI in a neighbourhood of zero, bi E C, and bk i= 0). Therefore a is a pole and the exponent k (which is uniquely determined) is called the multiplicity of the pole. By putting f(a) = 00, the function f becomes a holomorphic mapping of a neighbourhood of a into the Riemann sphere C with multiplicity k at a. Conversely, any such mapping must be of the form (m). Consequently: On anyone-dimensional manifold M, the singular points of a meromorphic function are always (isolated) poles. A complete meromorphic function on M is the restriction of a holomorphic mapping f: M - - - t C, f oj. 00, to the set {J i= oo} and vice versa. Then {J = oo} is the set of its poles, and the multiplicity of any of these poles, say a, is equal to the multiplicity of the mapping f at the point a.

p. 88). The meromorphic functions on !vI are usually defined as the sections (on M) of this sheaf. They are the mappings 'P: M --+ 9J1 such that for each a E M there is a meromorphic fraction g/ h on an open neighbourhood U of the point a, such that 'P(z) = gz/hz E Mz for z E U. They constitute a ring 9J1 M containing OM as a subring (after the identification OM '3 h --+ (z --+ h,) E 9J1 M ). Now, the meromorphic functions defined as the sections of the sheaf 9J1 correspond precisely to the complete meromorphic functions defined above. Indeed, it is easy to see that the mapping MM '3 f --+ (z --+ flz) E 9Jl M is an OM-isomorphism of rings. Finally, meromorphic functions can be regarded as the equivalence classes of the meromorphic elements of

O:U.

(130) It can happen that the field of fractions of the ring OM is a proper subset of the field MM· Indeed, if M is compact, then OM is the ring of constants (because of the maximum principle; see C. 3.9). On the other hand, on any multiprojective space of positive dimension there are non-constant meromorphic functions (see nO 7 below).

442

VII. Analyticity and algebraicity

VII.14.4

=

=

Consider a one-dimensional complex torus T. Then T CIA, where A Za + Zb is a lattice on C (see C. 3.21). The natural homomorphism 7r: C ........... T is a doubly-periodic mapping with periods a and b. To every doubly-periodic mapping f: C ........... N (with periods a, b and values in a set N) corresponds in a one-to-one way a (unique) mapping

j: T ........... N such that j 0 7r = f. Since 7r is a locally biholomorphic surjection, it follows (see C. 4.2) that if N is a manifold, then the above bijection establishes a one-to-one correspondence between the holomorphic doubly-periodic mappings of C into N (with periods a and b) and the holomorphic mappings of T into N. In particular, if N C, then the meromorphic functions on the torus T correspond to the elliptic functions on C (with periods a and b) ( 31 ).

=

The residue theorem for the logarithmic derivative on a suitably chosen rectangle of periodicity c + [0, l]a + [0, Ijb implies (see [5], IX. 4) that every non-constant meromorphic function on the torus T (such a function must have a pole, by the maximum principle; see C. 3.9) attains each value exactly r times (counted with multiplicities), where r is the sum of the multiplicities of the poles of the function. Consider the Weierstrass elliptic function

p(z)

= z-2 +

L

(z - c)-2

e32 ).

cEA\O

The function is even. Its derivative q(z) = p'(Z) = -2 L:A (z - c)-3 is an odd elliptic function. Let p and g be the corresponding meromorphic functions on the torus T. Each has a unique pole at 0 (of multiplicities two and three, respectively). In particular, p attains each value at two points. Moreover, p( -C) = p(C) and g( -C) = -g(C). The torus T can be embedded into the projective space P2 via the mapping the formula

g(C) = (p(C) , g(C) E C 2 for (

=I

0

and

g:

T

--+

P2 = C2, defined by

g(O) = wE C2 ,

where the homogeneous coordinates of the point ware (0,0, -2). Indeed, g is an injection. To see this, let g(C) gee'). If ( 0, then (' O. Let ( =I O. Then P(C) g((') and q(C) = q(/); if, in addition, q(C) = 0, then the multiplicity of p at ( is :2: 2, and so (' = (. On the other hand, if g(C) =I 0, then g(C) =I g(-C), but p(C) = pc-C), hence

=

=

=

= ('.

=

=

we must have ( Now, 9 corresponds to the mapping g: C ........... P 2 C2 defined by g(z) = (p(z), q(z»), for z E C \ A, and g(A) = w. It is enough to prove that g is an immersion, since then the mapping 9 is also an immersion and, as it is simultaneously a homeomorphism onto its range, it must be an embedding (see C. 3.14). Now, if z Y'. A, then one cannot have pi (z) = q' (z) = 0, for if this were true, the multiplicity of p at z would be :2: 3, which is impossible. Furthermore, for z =I 0 in a neighbourhood of zero, we have p(z) z-2 + a(z) and q(z) -2z- 3 + a/(z), where a is a holomorphic function. Therefore, in a neighbourhood of zero, (z3, z + z3 a (z), -2 + z3 a /(z») are homogeneous coordinates of g(z), and thus (by taking the 2nd canonical chart) we conclude that g is an immersion at O. Thus, by double-periodicity, g is an immersion at each point of the lattice A.

=

(

31

)

=

i.e., to the doubly-periodic meromorphic functions. (See [5], IX. 4).

(132) For every A' C A, the sum L:A'(z - c)-2 is almost uniformly convergent in C \ A', because L:A \0

Icl- 3 < 00.

Double-periodicity of the function p is a direct consequence of

the fact that the function is even and its derivative is doubly-periodic.

443

§14. Meromorphic functions and rational functions

Thus the image W == g(T) C P2 is a submanifold that is biholomorphic to the one-dimensional (complex) torus. By Chow's theorem (see 6.1) it is algebraic, and hence Wo == wnC 2 == g(T\w) == g(C\A) = {(p(t), q(t) : t E C\A} C C 2 is a principal algebraic set (see 6.3; and 11.3, corollary 1). Its degree is :S 3, because (see 11.5, proposition 7) the line {O'z + {Jw + 'Y == O} intersects the set in at most three points, since the meromorphic function O'p + (Jij + 'Y has only one pole, its multiplicity is :S 3, and so the function vanishes at most at three points. Thus Wo == V(P), where P is a non-zero polynomial of degree :S 3, and we have the relation p(p(z),p'(z)) = 0 (133).

4. We will now prove a theorem that characterizes meromorphic functions.

THEOREM 1. Let f: M \ Z ------- C be a holomorphic function, where Z C jV! is a nowhere dense analytic set. The following conditions are equivalent:

(1) the function f is meromorphic on M; (2) the closure! in M x C is analytic; (3) the closure! in M x C is analytic; (4) the graph of f is an analytically constructible set in M

X

C;

(5) the graph of f is an analytically constructible set in M

X

C;

(6) the pair consisting of the closure! in M

X

C and the set M

the condition of regular separation;

X

= satisfies

(7) the function f satisfies the condition (r) near Z. PROOF. The implication (1) ::::::::} (7) follows from the proposition in IV. 7.2 (134). Indeed, take a E Z. There exists a coordinate system cp : G ------- U at a and a meromorphic fraction g/h on U containing f 0 cp-l (see nO 2). Then cp(Z) ::J V(h) = h-1(0), and (as 0 E cp(Z)) we have Ih(OI >

Cfl((,CP(Z))' in the neighbourhood cp(~) of zero in cm. Here ~ eGis a compact neighbourhood of a, and

Ig(O/h(OI

~ ]{fl((,CP(Z))-S for ( E

means that If(z)1 ~ ]{ (}",(z, Z

> 0, s > O. Hence If(cp-l(O)1 = cp(~) \ cp(Z) with some]{ > 0, which C

n G)-S for z E

~ \

Z. (See 3.6.)

The conditions (3), (6), and (7) are equivalent, according to proposition 2 from 7.3, whereas the conditions (2), (4), as well as (3), (5) are equivalent (133) It can be shown (see (5), IX. 5) that (p')2 - 4p 3 + O'p + {J == 0 for some 0', {J E C that depend on A and are such that 0'3 - 27{J2 ¥ o. The converse is also true. For any such 0', {J, there is a lattice A for which the Weierstrass function satisfies the above equation.

Since the polynomial w 2 - 4z 3 + az + {J is irreducible, it follows that the closure in C2 of the algebraic set {w 2 - 4z 3 + O'Z + {J = O} is a submanifold which is biholomorphic to a one-dimensional torus, provided that a 3 - 27{J2 ¥ O.

(134) See also the remark following the proposition.

444

VII.14.5

VII. Analyticity and algebraicity

because of proposition 5 from IV. 8.3. The implication (3) ===>- (2) is trivial. If the condition (2) is satisfied, then the closure f in M x C is analytic in (M X C) \ (Z X (Xl) (135) and of constant dimension m (see IV. 2.5). Therefore, since dim(Z x (Xl) < m, condition (3) follows by the Remmert-Stein theorem (see IV. 6.3). Thus we have shown the equivalence of the conditions (2)-(7). To finish the proof, it suffices to prove the implication (3) ===>- (1). Let a E Z. The closure fin M x C, where C = PI = P(C 2 ), is analytic of constant dimension m (see IV. 2.5). Therefore, in view of the proposition from 6.5, there is an open connected neighbourhood U of the point a such that fu is defined by a holomorphic C 2 -homogeneous function in U x C 2 . That is (see

C. 3.18), it is defined by a function of the form F(z, t, u) = I:~ bv(z)tk-vu v , where r ::; k, the coefficients bv are holomorphic in U, and br O. Thus, if z E U \ Z and br(z) =1= 0, then J(z) is the only root of the equation I:~ bv(z)u V = 0 (see 3.1, the identification C '-7 Pd, so we must have r > 0 and br-1(z) = -rbr(z)J(z). Therefore Ju ~ -br_I/rb r .

t

If J E OM' then also (3) ===>- (1), according to the last part of the proof. Furthermore (1) ===>- (3), because if the function J is meromorphic, then so is j and the graphs of both functions have the same closure in M x C (see nO 1). Theorem 1, combined with the analytic graph theorem (see V. 1, corollary 3 from theorem 2) and proposition 5 from IV. 8.3, implies the following THEOREM 1'. Let f be a continuous function on an open dense subset of the manifold M. Then the following conditions are equivalent:

(1) the function f is meromorphic on Mj (2) the closure

f

in M x C is analytic.

Under the assumption that the domain of f is analytically constructible, the above conditions are equivalent to each of the following ones: (3) the closure

f

in M x C is analyticj

(4) the graph of f is an analytically constructible set in M x Cj (5) the graph of f is an analytically constructible set in M x

C.

COROLLARY. If f is a meromorphic function on M and a is an indeterminate point of f, then (f)a = C, where f is the closure in M X C.

Indeed, for each z E M we have #(f)z < (Xl or (f)z = C, and the set {z EM: (f)z = C} is closed. Therefore, if we had #(f)a < (Xl, then for some open neighbourhood U of the point a the natural projection 7r: Ju ---+ U would have finite fibres. Since the analytic set (135) Since the set

J n «(M \

Z) x C)

=f

f

is of constant dimension m

is analytic in (M \ Z) x

C.

445

§14. Meromorphic functions and rational functions

(see IV. 2.5), the projection 7r would be open (see V. 7.1, proposition 1) and the function U :3 z ---+ #(f)z would be lower semi continuous (see B. 2.1). But #(f)a > 1, and so we would have #(f)z > 1 in a neighbourhood of the point a, which is impossible. Consequently, (f)a = C. 5. We say that the functions h, ... , /k E OM are analytically dependent if rank(h, ... ,fk) < k. We say that they are algebraically dependent if P(h, ... , !k) = 0 for some non-zero polynomial P E Pk (136).

THE SIEGEL-THIMM THEOREM. Assume that the manifold At is compact. Then meromorphic functions h, ... ,fk on M are algebraically dependent if and only if they are analytically dependent. PROOF (137). One may assume that the functions fi have a common domain G and that the complement of G is analytic (138). Let f = (h,···, fk) : G ---+ C k . By theorem 1 from n° 4, the sets fi C M x C are analytically constructible and so is the set f C M X C k . According to the Chevalley-Remmert theorem (see V. 5.1), the set f(G) = 7r(f), where 7r: M X C k ---+ C k is the natural projection, is constructible in Ck(see 8.3, proposition 1). Since the algebraic dependence of the functions h, ... ,!k means that there exists a non-zero polynomial from Pk that vanishes on f( G), it is equivalent to the condition intf(G) = 0 (see 8.1 and 8.3, proposition 2). Therefore it is equivalent to the analytic dependence of the functions h, ... ,fk (see V. 1 theorem 1, and C. 4.2 (139)). COROLLARY. Under the hypotheses of the theorem, if k meromorphic functions h, ... ,fk are algebraically dependent.

>

Tn,

then the

Let V be an irreducible algebraic subset of a vector space M. Then the associated ring R(V) is an integral domain (see 12.1) and its field offractions K(V) - called the field of rational functions on V - is an extension of the field C. We will prove that: dim V is equal to the transcendence degree of the field K(V) over C, i.e., to the supremum of the number of elements of K(V) which are algebraically independent over C (140).

e

36 ) Naturally, (h, ... ,!k) denotes here the diagonal product of the restrictions to the intersection of the domains of h, ... , !k. In each of the conditions, the functions J; can be replaced by equivalent ones (see C. 3.12).

(137) See Narasimhan [33), p. 135. (138) See footnote (136), and nO 1 and 2. (139) If h, ... , fk are not analytically dependent, then the mapping f is a submersion at some point of the set G.

(140) Elements

Xl, ... , Xr

of an extension L' of a field L are said to be algebraically depen-

446

VIl.14.6

VII. Analyticity and algebraicity

Indeed, take m elements of the field K(V). They are of the form it I g, ... . . . , f mig, where fi E R(V), 9 E R(V) \ O. The graph of the mapping {g

i- O}

3 z ~ (it(z)lg(z), ... , fm(z)1 f(z)) E

em

is constructible, so, by the Chevalley theorem (see 8.3), the range H c em of the mapping is constructible. Now, the algebraic independence of the elements it I g, ... , f mig means, as can be easily checked (141), that there is a polynomial P E Pm \0 such that p(it (z)1 g(z), ... ,fm(z)1 g(z)) = 0 in the set {g i- OJ, i.e., it vanishes on H. This is equivalent to the condition dimH < m (because dimH = dimH; see 8.1 and 8.3 proposition 2). Set k = dim V. We have dimH ::; k (see V. 1, corollary 1 of theorem 1), which means that any m elements of K(V), with m > k, must be algebraically dependent. On the other hand, by taking 9 = 1 and J; = (Fi) v, where F1 , . .. ,Fk E M* are such that V(Fl"" ,Fk) is a linear complement of the tangent space to the k-dimensional manifold VO at one of its points, we get dimH = k (since (it, ... , !k) v is an immersion at that point). This means that it, ... ,!k are algebraically independent. 6. In what follows, let M be an m-dimensional vector space.

By a rational function on M we mean any function {h i- O} 3 z ~ g(z)lh(z) E e, where g, hE P(M) and h i- O. Such a function is a meromor-

phic fraction 9 I h on M, and hence it is a meromorphic function on M. M x

Obviously, the graph of a rational function on M is constructible in

e

(142).

We will prove the following two propositions. PROPOSITION 1. The complete rational functions on M are precisely the meromorphic fractions glh on M, where g, hE P(M) are relatively prime. PROPOSITION

2. If f is a rational function on M, then so is

j.

Indeed, if f is a meromorphic fraction of the form glh, where g, h E P(M) are relatively prime, then the set of removable singular points of the function f is open in V(h), and hence it is of constant dimension (m - 1); it is contained in the algebraic set V(g, h) of dimension::; m - 2. (See 11.3, corollaries 1 and 3 of proposition 5.) Therefore it must be empty, and thus dent over L if P(Xl, ... ,x r

)

= 0 for some P E L[Xl'.'. ,XrJ \ O.

(141) Using the fact that T'p(ZI, . .. , Zm) = Q(T ZI, ... ,TZm) for some sEN and Q E C[T,Yl, ... , Ym].

(142) Since it is the set {g(z) - th(z) = 0, h(z) ;f. O}.

447

§14. Meromorphic functions and rational functions

the function J is complete. Next, any rational function J on M is equivalent (owing to (*) from n O 2) to a meromorphic fraction 9 / h, where g, h E P( M) are relatively prime; hence 9 / h must be complete, and so j = 9 / h. If, in addition, the function J is complete, then J = g/h (see nO 1). It is easy to verify that the set of complete rational functions on M with addition

and multiplication given by (I, g) ---t I+g and (I, g) ---t jg, respectively, is a field (143) that is isomorphic to the field of fractions of the ring P(M). (The mapping that takes a meromorphic fraction g/h, where g, hE P(M) are relatively prime, to the element g/h of the field of fractions of the ring P(M), is an isomorphism.)

7. Now, let N = Xl X '" vector spaces. Set n = dim N.

X

X k and

N = Xl

X ... X

Xk,

where Xi are

THE HURWITZ THEOREM. IJ ZeN is an algebraic set oj constant dimension n - 1, then Jor any function J: N \ Z - - t C we have the equivalence:

(J is rational on N)

PROOF. If the function in

Nx C

(J is meromorphic on N)

J is rational on N, then its graph is constructible

(see n° 6; and 8.3, proposition 1). By theorem 1 (condition (5)),

the function function

{==:}

J is meromorphic on N

J is meromorphic on N.

(144). Conversely, suppose that the

According to theorem 1 (condition (3)) and

Chow's theorem from 6.4, the closure J in N x C is algebraic of constant dimension n (see IV. 2.5), and so is its trace J n (N x C) in N x C (see 6.4). It follows (see 11.3, corollary 1 from proposition 5) that it is equal to V(F), where F(z, t) = ~~ av(z)t", a v E P(M), and a r # O. Hence, if z E N\Z and ar(z) # 0, then J(z) is the only root of the equation ~~ a,,(z)t" = 0, and so we must have r > 0 and ar-l(z) = -rar(z)J(z). Thus J c::' -ar-I!ra r (l45) and so J C g/h, where g, h E P(M), h # 0 (see proposition 2 from n° 6). Since Z = V(d), where d E P(N) \ 0 (see 11.3, corollary 1 of proposition 5), we have J = gd/hd. This means that the function J is rational on N. THEOREM 2. If Z C M is an algebraic set of constant dimension (m -1), then for a function J: M \ Z - - t C the following conditions are equivalent: (1) the function f is rational on Mj (143) It is clearly a su bring of the ring MM. (144) It can be verified directly by using the charts of the manifold N (see (**) from 3.5). (l45) The function f is also meromorphic on N (see nO 2).

448

VII.l4.8

VII. Analyticity and algebraicity

(2) the function f is continuous and its graph is constructible in M x C; (3) the function f is holomorphic and

where

g = minCe, 1)

(after endowing M with a norm and putting e( z, 0)

=

1.) The implication (1) ===} (2) is trivial. If the condition (2) is satisfied, then (see 8.3, proposition 1; and IV. 8.3, proposition 5) the closure 1 in M x C is analytic. Therefore, in view of theorem l' (condition (2)) combined with the Hurwitz theorem, the function f is rational on M. The equivalence (1) ~ (3) is implied by the following easy-to-check inequalities for polynomial hE P(M) of degree k ~ 0: PROOF.

where c, C > 0 and Z = V(h) (146). Indeed, if f = gjh, where g, hE P(M) and h ¥ 0, then Z = V(h) and the condition (#) follows from the first inequality. On the other hand, assuming the condition (#), since Z = V(J) for some h E P(M) \ 0 (see 11.3, corollary 1 of proposition 5), the second inequality yields If(z)h(z)PI ::::; KCP(l + Izi)P+kp in M \ Z. Therefore, by Liouville's theorem, fh P is the restriction of a polynomial g E P(M) (see C. 1.8 and II. 3.5), and so f = gjh P • One can prove that the condition (#) (for a locally bounded function f on M \ Z) is equivalent to the condition (r) on M near the set Z U Moo = Z U Moo. This last set is algebraic in M (see 6.3) of constant dimension m - 1. Then the equivalence (1) (3) is a consequence of Hurwitz's theorem and theorem 1 (condition (7)). COROLLARY. A continuou.s function f: M structible in M x C is a polynomial.

---t

C whose graph is con-

THEOREM 3. Every continuous function whose graph f c M x C is con.structible, and whose domain is dense, is the restriction of a rational function on M ( 47 ) .

(146) As for the first inequality, one may assume that M = em and h is a monic polynomial in Zm (see A. 3.3). Then, if Z E M, we have \h(z)\ = \Z-(l\ . . . \Z-(k\. where (1, ... , (k E Z. Hence \h( z) \ 2: /?( z, Z)k. The second inequality is a simple consequence of the mean-value theorem. (

47 ) Thus, it is enough to assume that the domain of f is open. Without any assumption on the domain of t, the theorem is no longer true (see nO 8 below, the remark following theorem 4).

§14. Meromorphic functions and rational functions

449

PROOF. By the Chevalley theorem (see 8.3), the closure Z ofthe complement of the domain of 1 is algebraic and nowhere dense (see 8.3, proposition 2; and IV. 8.3, lemma 2). One may assume that 1: M \ Z --+ C (because 1M\Z is dense in 1 (148»). Now, Z = ZOUZl, where Zo is algebraic of constant dimension (m - 1) while Zl is algebraic of dimension < m - l. The graph of 1 is closed in (M \ Z) x C and hence analytic (see IV. 8.3, proposition 5). Therefore, in view of the analytic graph theorem (see V. I, corollary 3 of theorem 2), the function 1 is holomorphic in M \ Z. By the Hartogs theorem (see III. 4.2), it has a holomorphic extension extension is equal to

f n ((M \

and so it is constructible in M

2, the function

j

j

on M \ Zo. The graph of this

Zo) x C), where

f

is the closure in M x C,

x C (see 8.3, proposition 2). Thus, by theorem

is rational on M.

8. Now we will prove the following THEOREM 4 (ZARISKI'S CONSTRUCTIBLE GRAPH THEOREM). For each function 1: S --+ C with domain SCM and graph constructible in IVfx C (149), there exists a quasi-algebraic set T C S which is open and dense in S, and such that fT is the restriction of a rational function on M, ~.e.,

1(z)

=

g(z)jh(z) and h(z) -=I- 0

for z E T ,

where g, hE P(M). REMARK. The function 1 itself, even if it is continuous, does not have to be the restriction of a rational function. This is the case, because there are continuous functions defined on an algebraic set and with algebraic graph which are not even holomorphic. (See the example from V. 8.1 (150).) In the proof of the theorem we will need the following four lemmas: LEMMA l. Let V be an algebraic subset of the Cartesian product X x Y of vector 3paces which is non-empty and of constant dimension equal to dim X. Suppose that the natural projection 7f: V --+ X is proper. Then there exist a nowhere dense algebraic set Z C X and p > 0 such that # Vx = p for x E X \ Z and

X \ Z

=

{x: y E Vx

==}

V is a topographic submani10ld at (x,y)} .

(148) We take a complete rational extension of fM\Z. See nO 1,2, and 6. (149) Then S is also constructible.

(150) The function in that example is equal to

w/z

outside the point (0,0).

450

VII. Analyticity and algebraicity

VII.l4.8

The lemma is contained in the Andreotti-Stoll theorem (see V. 7.2 and C 3.17; the projection 7r is then a p-sheeted *-covering whose exceptional set is Z = {x: # Vx < p}). One only needs to check that Z is algebraic. But this is true, because Z is constructible (see 8.3, proposition 2): the complement of Z is the image by the projection onto X of the constructible set

{(X,Yl,··.,Yp) E X x

YP:

(x,Yd, ... ,(x,yp) E V, Yl, ... ,Yn are distinct}.

LEMMA 2. Let 0 f V* M be an irreducible algebraic set, and let Z be a nowhere dense algebraic subset of V. Let f: V \ Z ----+ C be a continuous function with constructible graph in M X C, and let E c M \ V be a finite

set. Then there are: an irreducible algebraic set

11

:J V of codimension 1

and disjoint from E, its nowhere dense algebraic subset Z :J Z u 11*, and a continuous function j: 11 \ Z ----+ C with constructible graph in M x C, such that V \ Z f 0 and j = f in V \ Z . PROOF. The ideal I = I(V) is prime (see 1.6) and V = V(I) (see 1.2). One may assume that M = en and that the ideal I is k-regular (see 9.1). Furthermore, (using the notation from §9) one may assume, in view of the proposition from 9.3 (and because of 9.2), that the hypotheses of Ruckert's lemma from 9.3 are satisfied, and

(#) Then (using the notation from Ruckert's lemma) we have

(R)

V{8#O} =

{(u,v): t5(u) i=- 0, PHl(U,ZHd = 0, t5(u)Zj=Qj(U,Zk+I), j=k+2, ... ,n}.

Therefore the set V is k-dimensional (see §10, corollary 2; and 8.2). The polynomial PHI is irreducible in PHI, and hence also in P n (see A. 2.3). Thus the algebraic set

(##) is irreducible of codimension 1 (see 11.2, corollary 1 and (3)). As PHI E I (see 9.3, (*)), so V

1i'(V)

= 1i'(11).

c V.

Next, according to corollary 2 from §10, we have

Consequently, it follows from (#) that

11 n E = 0.

Set

§14. Meromorphic functions and rational functions

By

(##),

we have V{6~O} C yO, and so

Z :J

Z U V*. Since dim(Z n V) < k

(see IV. 2.8, proposition 3; and then II. 1.4 and 9.3 (**)), we have V \ (which implies that

451

Z is nowhere dense in V).

Z i- 0

Finally, the function

is continuous with constructible graph (see 8.3, the corollary of Chevalley's theorem), and if z E V \

and so l(z)

Z,

then, in view of (R), we have

= fez).

LEMMA 3. If Z c M is an algebraic set of codimension 2: 2 and if a E M, then the set {,\ E P(M): (a + A) n Z c a} contains an open dense subset of

P(M). Indeed, one may assume that a = 0 and then the complement of our set, being equal to the image of the set Z \ 0 under the mapping a = atE, is constructible (see 8.1). That is because the graph of a is constructible in ill x P( M) (see 8.7). The dimension of that complement is ~ dim Z < dimP(.l\!J) (see V. 1, corollary 1 of theorem 1), and thus its closure is nowhere dense in P(M) (see IV. 8.5 and II. 1.2). LEMMA

4. If

~,v

c M

is an algebraic set of constant dimension (n - 1)

and if a E vV o , then the set

{,\ E P(M): a

+ A intersects

W transversally}

contains an open dense subset of P(.M). PROOF. One may assume that a = O. The set T = an (W x P(M)) constructible in M x P(M) (see 8.7; and 8.3, proposition 1) and it is analytic in (M \ 0) x P(M). Being equal to the image of the set W \ 0 under the biholomorphic mapping (3: M \ 0 :') z ~ (z, Cz) E a, it is of constant dimension n -1. The set T c M x P( M) is algebraic and of constant dimension n - 1 (see IV. 8.3, proposition 5; and IV. 8.5). Furthermore, the natural projection 7f: T ~ P(M) is holomorphic and proper. Therefore - in view of proposition 3 from V. 7.3 - there is an open dense subset n c P(M) such that 7f is biholomorphic at the point (z, A) E T if A E n, and hence whenever A E nand z E An W \ O. Now, aW\O = 7f 0 (3', where (3' = (3W\O : IS

W \ 0 ~ T is biholomorphic. Let A E nand z E An (W \ 0). Therefore aW\O is biholomorphic at z, and hence z E W O (see V. 3.4, the proposition)

452

VII.l4.8

VII. Analyticity and algebraicity

and the differential dzcxw\o = (dzCX)TzW is an isomorphism. Since cx = A on A \ 0, so A E ker dzcx (see C. 3.11), and hence A ct. Tz W. Thus A intersects W transversally at z. Finally, it is enough to observe that the set of lines A E P(M) that intersect W transversally at 0 (i.e., the set {A: A ct. To W}) is open and dense. Let us prove the lemma in a more direct way. We may assume that M =

cn

and a = O.

=

First suppose that W is irreducible. Then W Yep), where P E P(M) is an irreducible polynomial (see 11.3, corollary 1 of proposition 5). Consider the polynomial

Q=

(###)

~ 8P Zi.

L...1

8Zi

If Q = 0 on W, then Q is divisible by P (see 11.2, proposition 4), but the degree of Q is not greater than that of P, and so one must have Q = aP, where a E Cj then P is a homogeneous polynomial (1S1), and hence W is a cone. Thus each line of the set P(M) \ W-, which is open and dense (since dim Wn - 2j see 6.2), can intersect Wonly at O. On the other hand, if V(Q) 1J W, then the co dimension of the set W n V(Q) is ~ 2 (see IV. 2.8, proposition 3). Therefore, by lemma 3, every line from an open dense subset ofP(M) can intersect the set WnV(Q) only at 0, i.e., it can intersect W\O only at points Z for which Q(z) I: O. Hence, in view of (###), it intersects only at regular points of W, transversally at each of those points.

=

In the general case, let W = WI U ... U lVr be the decomposition into simple components. Then the codimension of the set W' Ui,ti Wi nWj is ~ 2 (see IV. 2.8, proposition

=

3), and according to lemma 3, each line from an open dense subset of P(M) can intersect W' only at O. Consequently, each line from an open dense subset of P(M) intersects the set W \ 0 transversally. Finally, it is enough to observe that the set of lines from P(M) that intersects W transversally at 0 is open and dense. PROOF of theorem 4. One may assume that the function f is continuous, S = V \ Z, where Z eVe M are algebraic sets with Z nowhere dense in V (see 8.6, proposition 3; and 8.3, corollary 1 of proposition 2), and, in view of theorem 3 from nO 7, that 0 iM. So it is sufficient to prove that there exists a nowhere dense algebraic subset Z' ~ Z of V such that fv\Z' is the restriction of a rational function on M.

V*

Let V Take Ci E

=

VI U ... U Vr be the decomposition into simple components. Uj#i Vi, i = 1, ... , r. For each i = 1, ... , r, the set Z n Vi is nowhere dense in Vi (see IV. 2.9, corollary 1 of theorem 4). Accordingly,

Vi \

lemma 2 implies the existence of: an irreducible algebraic subset codimension 1 such that

(1)

Cj

It' if;

V;

~ V of

for j -::j:. i ,

(1S1) Because if a I: 0 and z E M, then the polynomial J(t) = P(tz) satisfies the condition tf'(t) = aJ(t), which implies (e.g., by comparing coefficients) that J(t) = eta for some c E C. Therefore P(tz) = t a P(z) for t E C, z E M.

§14. Meromorphic functions and rational functions

453

its nowhere dense algebraic set

(2) and a continuous function that

(3)

Vi \

ij: Vi \ Zj

Zj i- 0

and

-

-

~ C with constructible graph such

ij = f

III

Vi \ Zi .

The set

-

V=V1U ... UVr

(4)

is of constant dimension n - 1 and (4) is its decomposition into irreducible components (due to (1)). Therefore

(5)

Z=

UZj U UVi n Vj is nowhere dense in V and of co dimension

;::: 2

i¥j

(see B. 1; and IV. 2.8, proposition 3). Moreover, in view of (2),

(6)

Z =:l

Z U V*

(see IV. 2.9, theorem 4). Now take aj E Vi \

Z,

i = 1, ...

,T

(see (3)). Then, because of (6), we

have ai E V , and so lemma 4 implies that for each line A from an open dense subset of PCM) we have O

ai

+ A intersect V transversally,

i

=

1, ...

,T .

By lemma 3 and (5), each line A from an open dense subset of P(J'H) satisfies the conditions (ai

+ A) n Z = 0,

that is,

aiEVi\(Z+A), i=l, ... ,T. Finally, for each line A from an open dense set in P(M)

(*** )

A is a Sadullaev line for

(see 7.1, Sadullaev's theorem).

V

454

VII. Analyticity and algebraicity

VII.14.8

Consequently, there exists a line A E P(M) satisfying the conditions (*), (**) and (***). One may assume that M = N X C, where N is an (n -1)dimensional vector space, and that A = 0 X C. Let 7r: M ---+ N be the natural projection. Then (see 7.1) the projection 7r v : V ---+ N is proper, and lemma 1 yields the existence of: a nowhere dense algebraic set Eo C N and p > 0 such that (7)

# Vx

(8)N\Eo = {x EN: Y E

Vx

= p

===}

V

x E N \ Eo,

for

is a topographic submanifold at (x,y)}.

Therefore, owing to (*) and the fact that a;

+ A = 7r( a;) X C,

we obtain (see

C. 3.17)

(9)

7r( ai) E N \ Eo,

i

= 1, ... ,r

.

V \ Z = UCV; \ Z),

Now, in view of (5), we have the disjoint union

V \ Z (in view of (6)j see IV. 2.9, consequence, the function j: V \ z ---+ C defined by

of which are open in

j = j;

in

V; \ Z,

i

=

the terms

theorem 4). As a

1, ... ,r ,

is continuous and its graph is constructible. Moreover,

j =f

(10) because, by (3), we have

j = j; = f

V \

in in

Z

(V; \Z)n(Vi \Zi) = Vi \Z, i =

1, ... , r.

Consider now the Lagrange interpolation polynomial

L(Y1,'rJ1i.·. iYp,'rJpiY)

p ~

= ~ II ---'rJs , Y-Yi

s=l io/-s Ys - Yi

where Y1, 'rJ1, ... , YP' 'rJP' Y E C and Y1, ... ,Yp are distinct. Define

(11)

E

= Eo U 7r( Z) .

(It is algebraic and nowhere dense, by (5), since the projection 7r v is closed.) Then the set

iI =

{(x, Y1, ... , YP' y, z) E Nx Cp+2 : z

= L(Y1,i(X, Y1)j ... j Yp,i(x, Yp)j y),

(x,yd, ... ,(X,Yp)EV, XEN\E, Yl, ... ,Yp aredistinct}

455

§14. Meromorphic functions and rational functions

IS constructible

(152), and hence so is its image H C 1111 x C under the

projection N

X

Cp+2:1 (X,YI,'" ,yp,y,z)

---+

(x,y,z) EM xC.

Now, according to (7), (8) , and (11), if x E N \ L;, then #Vx

=p

and

V

is a topographic submanifold at each point (x, y) E if. Hence (owing to the symmetry of L) the set H is the graph of a continuous function on the set (N \ L;) x C = M \ 7T- I (L;). By theorem 3 from nO 7, there are polynomials g, hE P(M) such that

H(z) = g(z)jh(z) and h(z)

(12)

#- 0 in the set

M \

7T-

I

(L;)

We claim that (13)

Vx

V\

(L;). Then x E N \ L; and, by (7), we have = {YI, ... ,Yp}· Therefore y = Ys for some s. But (X,YI,'" 'YP,y,z) E fI,

Indeed, let (x, y) E

7T-

I

where

z

=

L(YI,J(X, yr); ... ; yp,J(x, yp); y)

=

](x, Ys)

=

j(x, y)

Thus (x,y,z) E H, and so j(x,y) = H(x,y). Suppose now that Z'

= V n 7T- 1 (L;).

This is true in view of (9) and (**),

ai

Then

ai

E Vi \ Z' because of (11).

rt 7T- 1(L;o) and

ai

rt 7T-I(7T(Z)) =

Z + A. Hence the set Z' n Vi is nowhere dense in Vi, which implies (see IV. 2.9, corollary 1 of theorem 4) that the set Z' is nowhere dense in V. Next, by (11) and (6), we have Z' ::J V n.2 ::J Z. Therefore V \ z' c V \.2, but also V \ Z' c V \ 7T- 1 (L;), and so (10) and (13) yield that! = H in V \ Z'. Thus, by (12), the function !V\Z' is the restriction of a rational function on M. Let us come back to the example from nO 3. Every elliptic function f is of the form where

. E (X x T)* for 7r. Let

0 ( 73 ) , we have (see nO 1b) 01 = OI'(V'(1)) ='

I(V'(1)a, V) = I(AI' V) n ... nI(Ar, V), and moreover, I(Ai, V) are prime ideals in the ring 0 (see V. 4.9). LEMMA 2. The localization OI to a prime ideal I of the ring 0 is always a ring without nilpotent elements.

r

Indeed, it is enough to check that E II ===> f E II (see A. 11.2). Let rg = 0, f E 0, g E 0 \ I. Let Va = Al U ... U Ar be the decomposition into simple germs. Since the rings 0 Ai are integral domains (see IV. 4.3), for each i we have fAi = 0 or gAi = 0, and thus Ig = O. LEMMA 3. If II, ... ,Ir are ideals of the ring R, then

It is sufficient to show the inclusion :J for r = 2, i.e., that O(f n J) :J Of n OJ for ideals f, J c R. Let h E Of n OJ. Since 0 = R + mn for each n E N (see n° 1b), we have h - '{In, h - 'l/Jn E mn for some '{In E f, 'l/Jn E J. Thus '{In - 'l/Jn E nn n (1 - J), in view of proposition 1 from

nO lb. Therefore, by the Artin-Rees lemma (see A. 9.4a), there is k such that '{In - 'l/Jn E nn-k(1 - J) for n ;::: k. Then '{In - 'l/Jn = an - f3n, where an E nn-kf, f3n = nn-kJ. But Xn = '{In - an = 'l/Jn - f3n E In J, and so h = Xn + an + (h - '{In) E 0(1 n J) + m n- k (for n ;::: k). Hence, by the corollary from Krull's theorem (see A. 10.5), we conclude that h E 0(1 n J). for

LEMMA 4. If f is a primary ideal of the ring R and X E R \ rad f, then 0 we have the implication

1E

Ix

E 01

===> 1 E Of .

Indeed, let Ix E 01. Since 0 = R+ mn for each n (see nO 1b), we have E m n for some '{In E R. Hence

1 - '{In

(73) Because a E V'(I) (see nO Ib).

467

§16. Serre's algebraic graph theorem

by proposition 1 from nO Ib (in view of mn = Onn; see n° Ib). Therefore .), where A E P(L) is such that A 0 is a prime element of R(S). Then the cone S is factorial. PROOF. The mapping P(L) :3 F --+ F* E R(S), where F* = F 0 , is a C-monomorphism of rings because the set (S) :J L \ L; is dense in L. By our hypothesis, A* is a prime element of the ring R(S). Therefore A is irreducible. (For otherwise A* would be the product of two non-constant elements of R(S), and thus it would be reducible; see 12.2.) Note that

for each f E R(S) \ 0 there is a polynomial FE P(L) which is { not divisible by A and such that F* = fA: for some r EN.

(1)

Indeed, according to the proposition from V. 3.4, the restriction \{I is a homeomorphism, and so the function f 0 \{I-I: L \ L; --+ C is continuous, has a constructible graph (see 8.3, the corollary of Chevalley's theorem), and is non-zero (since the set S\ -I(L;) is dense in S; see IV. 2.8, proposition 3). Therefore, by Zariski's constructible graph theorem (theorem 4 in 14.8; see also 11.2, proposition 4 with corollary 4), we have f 0 \{I-I = F/ Ar in L \ L; for some r E N and some polynomial F E P(L) that is not divisible by A. Hence F* = fA: (because the set S \ -I(L;) is dense in S). Since the ring R( S) is a noetherian integral domain (see 12.1), it is enough to prove that each of its irreducible elements is prime (see A. 9.5). Let p E R(S) be an irreducible element. Omitting the trivial case, one may assume that p is not divisible by A*. Suppose that p is a divisor of the product of elements f, g E R(S) \ 0 (236). For some h E R(S) \ 0, we have

(2)

ph

=

fg .

It follows from (1) that there are P,H,F,G E P(L) and r,s EN such that P is not divisible by A and

Now the polynomial P is irreducible. Indeed, if it were not, then (since

P is non-constant) it would be equal to the product of non-invertible polynomials pl,P" E P(L) that are not divisible by A. Then pA: = P;P;', which

would imply (since A* is prime) that P; = p' A:', P;' = p" A:" , and p = p'p", where p',p" E R(S). Moreover, the elements p',p" would be non-invertible (for otherwise, see 12.2, one of them, e.g., p' would be equal to a constant c

f. 0,

and then pI = dr'). This would contradict the irreducibility of p.

(236) We omit the trivial case f9

= O.

500

VII. Analyticity and algebraicity

VII.19.3

The identities (2) and (3) yield that P*H*A! = F*G*, hence PHA' FG, and so, e.g., F is divisible by P, i.e., PQ = F for some Q E peL). In view of (3), we have pQ* = fA!, and therefore Q* is divisible by A! (because p is not divisible by A*). Thus p is a divisor of f. 3. Let X be an n-dimensional vector space. Consider the embedded Grassmannian

(where 1 ::::; k ::::; n) and its cone (i.e., the Grassmann cone)

9

= 9k(X)

= Gk(Xr = {Zl

A ... A

Zk : Zl, ... , Zk EX}

C

Ak X

(see 4.4). Let el, ... , en be a basis of the space X. Then {eo} oE!I., where eo e 01 A ... A e Ok is a basis of the space A k X. Take the polynomial mapping have

1{

= {J 0 x: f

n

E

peAk X)}

x:

C J :3 Z

(237) and

9

--+

.z= Po(Z)e o E A kX.

= X(C

J

),

1

1

for

!I.

We

because

n

(LZljej) A ... A (LZkjej) = LPo(Z)eo

=

Z = [Zij] E

cJ

(see A. 1.20). Now the ring homomorphism

whose range is H and whose kernel is the ideal I(9) induces the isomorphism

(1) On the other hand, we have the natural isomorphism

P(A k X)jI(9)

(2)

--+

R(9)

(see 12.1). As a result, we conclude that: The ring R(9) of the Grassmann cone is isomorphic to the Hodge ring

Consequently, we have the following version of (237) Because fox

=i

i

0 P, where denotes the polynomial f in the coordinate system (in A k X) corresponding to the basis {e" }"E!I.'

501

§19. The Andreotti-Salmon theorem THE

ANDREOTTI-SALMON

THEOREM.

Gk(X) C P(A k X) is factorial.

The

embedded

Grassmannian

We are going to give another proof of this theorem. Namely, by using the lemma from nO 2 we will show that the Grassmann cone 9 is factorial. Note first that 9 C Ak X is an irreducible algebraic cone. Indeed (see 6.2), the Grassmannian Gk(X) C p(AkX) is a connected submanifold (see 4.4 and B. 6.8), and hence it is an irreducible algebraic set (see IV. 2.8, proposition 2; Chow's theorem in 6.1; and proposition 3 in 11.1). Next, in view of Hodge's lemma, the restriction J.Lg of the form J.L: AkX32...::CaCa~Ce:EC, IS

where

E:=(l, ... ,k),

a prime element of the ring R(9). This is so because the images under

the isomorphisms (1) and (2) of the equivalence class of J.L E P(A k X) are J.L 0 X = Pe: and J.Lg, respectively. Set H

= ker

J.L. Then H

=

La,te: Cea. We have the equality n

(***) 9\H

=

{a(ci +vdt\·· .t\(Ck+Vk): a E C\O, VI,··· ,Vk E

2...:: Ce;}

.

k+I

Indeed, for any element z E X denote by z', z" its components z' E

L~ CCi and z" E L~+I CCi. Now, the inclusion :J is obvious. Conversely, let WI t\ ... t\ Wk rf- H. Define U = L~ CWi. We have WI t\ ... t\ Wk E W~ t\ ... t\ W~ + H, and hence w~ t\ ... t\ w~ =f. 0, which means that w~, ... ,w~ are linearly independent. It follows that the projection U 3 z ~ z' E

L~ CCi is an isomorphism. Consequently, the space U has a basis of the form el+vl, ... ,ek+vk,wherevl,···,Vk E L~+l CC s (238). ThuswIt\ ... t\Wk

=

a( e - 1 + VI) t\ ... t\ (Ck + Vk) for some a E C \ O.

We have the direct sum Ak X = L + N, where Land N are the subspaces generated by {Cal, .. ,ak : Cl'k-l ~ k} and {cal, ... ,ak : Cl'k-l ::::: k + I}, respectively. Accordingly, n

(###) L \ H

+ (2...:: bIjC j)

= {aci t\ ... t\ Ck

t\ e2 t\ ... t\ Ck

+ ...

k+l

n

... + cl

t\ ... t\

ek-l

t\ (2...:: k+l

(238) It corresponds to the basis

ej , . . . ,

q.

h jCj)

a E C \ 0, bij E C}.

502

VII. Analyticity and algebraicity

VII.20.1

Let 7r: Ak X - - - t L be the natural projection (with kernel N). The proof would be complete if we could apply the lemma from nO 2 to the regular mapping 7rg: 9 - - - t L and the polynomial JiL' Therefore it is enough to check that the hypotheses of the lemma are satisfied. Now, because of the inclusion N C H, we have

Hence JiL 0 7rg E R(9) is a prime element. Since V(/-lL) = H n L, it only remains to check that the mapping 7rg\H : 9 \ H ---t L \ H is bijective. But because of (***) and (###), this follows from the identity

n

=

n

aell\·· .I\ek +a(l.: Cljej) l\ e21\·· .I\q + ... +ell\·· .l\ek-1I\a(l.: Ckjej) k+l

k+l

for a, Cij E C.

§20. Chow's theorell1 on bihololl1orphic ll1appings of GraSSll1ann manifolds 1. In any Grassmann space Gk(X), where X is an n-dimensional vector

space and 1 :s k :s n, we define the relation of being adjacent (denoted by Y) by the formula UYV

{:::=::?

dim(U

n V) = k -

1

{:::=::?

dim(U

+ V) = k + 1 .

Observe that

Next, UYV

{:::=::?

(U

i= V

and p(U)

+ p(V) c

gk(X)) .

In fact, extend a basis Zl, ... , Zk-r of the subspace U n V to a basis Zl, . .. ... , Zk-r, UI, . .. , U r of the subspace U and to a basis Zl, ... , Zk-r, VI, . .. , VB of the subspace V. Then p(U) = Cu and p(V) = Cv, where U = ZI 1\ ... 1\

503

§20. Chow's theorem on biholomorphic mappings

Zk-rI\UII\ ... I\u r and v = zll\ ... l\zk- r I\VII\ ... I\vr' Now the condition UYV means exactly that r = 1. Thus the implication ==} is trivial. Conversely, suppose that the right hand side of (**) holds. Then r > 0 and U + v is simple. Therefore UI 1\ . .. 1\ Ur + VI 1\ ... 1\ Vr is simple and we must have r = 1 (see A. 1.21).

Let Z denote the class of sets Z C Gk(X) satisfying the condition: (L,N E Z,L =f. N) {=?- LYN. LEMMA 1. If 2 :S k :S n - 2, then the only maximal elements (with respect to inclusion) of the class Z are the sets of the form

PROOF.

The sets in the first family in (#) are maximal. For if N E

Gk(X) \ Sk(U), then NY L does not hold for L = U + Cv E Sk(U), where v E (X \ U) \ (Cu + N) and U E U \ N (because Cu + Cv eLand (Cu + Cv) n N = 0). This implies, in view of the property (*) and the equality (#) from B. 6.5, that the sets in the second family in (#) are also maximal.

N ow suppose that a set Z E Z is maximal. First note that

*

(Indeed, suppose that N 1J LI n L 2. Then N =f. L;, and so dim(N n Li) = k-l (i = 1,2). Next, NnL l =f. NnL 2 because, since NnL I nL 2 LI nL 2, we have dim(N n LI n L 2) < k - l. Hence N = N n Ll + N n L2 eLl + Ld Now take L 1 ,L 2 E Z, LI =f. L2 and set U = LI nL 2 , V = LI +L 2 . Clearly, it suffices to prove that Z C Sk(U) or Z C Gk(V). Suppose this is not so. Then there would exist L3 E Z\ Gk(V). Then L3 =f. LI and L3 =f. L 2 , but according to (##) one must have L3 :J U, and thus Ll n L3 = L2 n L3 = U (because of equality of the dimensions). Furthermore, there would exist NEZ \ Sk(U). Therefore, in view of (##), we would have N C ni dim U. 2. Let X and Y be n-dimensional vector spaces, and let 1 :S k :S n - 1. We have the following

e

39

)

Indeed, by taking a complementary line Ai to U in Li (i = 1,2,3), we get the direct

+ Al + A2 + A3, (since ni for

some E

Lo(AkX, Aky). Thus 1>(Gk(X)) c Gk(Y), and so (~h(X)) C ~h(Y) (see i- V and pX(U) + pX(V) c

B. 6.10). Now, if UYV, then by (**) we have U ~h(X). Therefore feU) i- f(V) and pY (f(U))

+ pY (f(V))

+ (pX (V)) = = q,(px(U) + pX(V)) c 9k(Y) = (pX (U))

.

So, by (**), we have feU)! f(V). LEMMA 3. Suppose that 2 :::; k :::; n - 2. If f: Gk(X) biholomorphic mapping; then we have the disjunction

--+

Gk(Y) zs a

(240) See (12bJ. (241) It is an isomorphism of algebraic spaces (see 17.13 or 17.14, corollary 2 of Chow's theorem).

(242) If n

= 2, then the mapping 7"1 is an isomorphism of projective spaces, by the theorem

from 13.5.

505

§20. Chow's theorem on biholomorphic mappings

(A) there exists a biholomorphic mapping g: Gk-l(X)

----t

Gk-l(Y) such

----t

Gk-l(Y) such

that

(1) or

(B) there exists a biholomorphic mapping h: Gk-l(X) that

PROOF. According to lemmas 1 and 2, the image under f of a set of the form (#) is always a set of the form (#). Now, if for some Uo E Gk-l (X) we have f(Sk(U o )) = Sk(U~), where U~ E Gk-l(Y), then for each U E

Gk-l(X) we have f(Sk(U)) = Sk(U ' ), where U' E Gk-l(Y). Indeed, this is so when UYUo, because then Sk(Uo) n Sk(U) = {Uo + U}. If we had f(Sk(U)) = Gk(V') with V' E Gk+l(X), then the set f(Sk(Uo ) n Sk(U)) = Sk(U~)nGk(V') would be infinite, which is impossible. Therefore it is enough to observe that for any U E Gk-1(X) which is different from Uo, we have UoYU 1Y ... YU. YU for some Ui E Gk-l (X). Clearly, in the remaining case,

for each U E Gk-l(X) we have f(Sk(U)) = Gk(V / ), where V' E Gk+l(X), Hence there exists a mapping 9 that satisfies the condition (1) or a mapping h that satisfies the condition (2). As both conditions imply injectivity (of the mapping 9 or h, respectively), it remains to show that the mapping is holomorphic (see V. 1, corollary 1 of theorem 2 43 )).

e

Let Uo E Gk-1(X). Take lines A,I-' C X such that the sum Uo + A + I-' is direct. Then (see B. 6.6) for each U from an open neighbourhood D of Uo in Gk(X), the sum U + A + I-' is direct, hence U + A U + I-' and feU + A) t feu + 1-'). But U + A, U + I-' E Sk(U). Hence, in case (A), we have

t

feU +A),f(U +1-') E Sk(g(U)), and so g(U) = feU +).) n feU

In case (B), we have feu

+ A),f(U + 1-')

h(U) = feU

e

----

+ 1-')

.

E Gk(h(U)), and thus

+ A) + feu + 1-')

.

43 ) Then the mapping 9 (or h, respectively) is proper and its range is closed, which implies that the mapping is surjective, since the Grassmannian space is connected.

506

VII.20.2

VII. Analyticity and algebraicity

In view of the proposition from 4.3, the above equalities show that the mapping g (or h, respectively) is holomorphic in D. REMARK. The case (B) can happen only when n = 2k. (This is so, because the dimensions of the manifolds Gk-l(X) and Gk+l(Y) are equal, which means that (k - l)(n - k + 1) = (k + l)(n - k - 1).) LEMMA 4. Assume that k

G k- 1(Y) and f: Gk(X)

2' 2. If FE Lo(X, Y), g = F(k-l) : Gk-l(X)

---+

-+

Gk(Y) is a bijection such that f(Sk(U)) =

Sk(g(U)) for U E Gk-l(X), then f = F(k)' Indeed, if T E Gk(X), then T = U + V for some U, V E G k- 1(X), and so F(T) = F(U) + F(V). Therefore f( {T}) = f(Sk(U) n Sk(V)) =

Sk(F(U)) n Sk(F(V)) = {F(T)}. PROOF of Chow's theorem. The case k = 1 is just the theorem from 13.5 (cf. also the remark in 18.2). Let 2 ~ k < ~n, and assume that the theorem is true for k - 1. By lemma 3 and the following remark, there exists a biholomorphic mapping g : Gk-l(X) ---+ Gk-l(Y) such that f(Sk(U)) =

Sk(g(U)) for U E G k- 1(X). Thus g = F(k-l), where F E Lo(X, Y). Hence, according to lemma 4, we have J = F(k). Therefore the theorem is true if 1~ k

< ~n.

Now, let ~n < k ~ n - 1. We have the biholomorphic mapping

Since 1 ~ n - k < ~n, the mapping is an isomorphism. Hence it is of the form

Ftn-k)' where FE Lo(X, Y). That is, we have rf 01- 10 (rr)-1 rf

0

(F(k»)-1

0

hn-

1

=

Ftn-k)

=

(see B. 6.9), and hence J = F(k)'

Finally, if n = 2k, we use lemma 3 again. In the case (A), we obtain = F(k) with some F E Lo(X, Y), as in the first part of the proof. In the case (B), there exists a biholomorphic mapping h: Gk-l(X) ---+ Gk+l(Y)

1

such that for U E Gk-1(X) we have J(Sk(U))

= Gk(h(U)).

This gives

(see B. 6.5, the equalities (#)). But since the mapping r[+1

h maps the space Gk-l(X) biholomorphically onto the space Gk-l(Y*), it is equal to F(k-l), where F E Lo(X, Y*). Consequently, in view of lemma 4, we have 0

r[ 0 J = F(k). This means that J = (r[)-l 0 F(k)' and the proof is complete.

REFERENCES [1] BALCERZYK S., J6ZEFIAK T., Commutative Rings (Polish), BM vol. 40, Warszawa 1985. [la] BIALYNICKI-BIRULA A., Algebra (Polish), BM vol. 47, Warszawa 1976. [2] BROWKIN J., Field Theory (Polish), BM vol. 49, Warszawa 1977. [3] LANG S., Algebra, Addison-Wesley, Reading Mass. 1965. [4] ATIYAH M.F., MACDONALD LG., Introduction to Commutative Algebra, Addison-Wesley, Reading Mass. 1969. [5] LEJA F., Complex Functions (Polish), BM vol. 29, Warszawa 1971. [6] SICIAK J., Introduction to the Theory of A nalytic Functions of Several Variables (Polish), an appendix in the book [5]. [7] ZARISKI 0., Samuel P., Commutative Algebra, Van Nostrand, Princeton 1958-1960.

*** [7a] ABHYANKAR S.S., Local Analytic Geometry, Academic Press, New York 1964. [7aa] ABHYANKAR S.S., Concepts of order and rank on a complex space, and a condition for normality, Math. Ann. 141 (1960), pp. 171-192. [7b] ANDREOTTI A., SALMON P., Anelli con unica decomponibilita in fattori primi, Monatsh. f. Math. 61 (1957), pp. 97-142. [8] ANDREOTTI A., STOLL W., Analytic and Algebraic Dependence of Meromorphic Functions, Lecture Notes in Math. 234, Springer, Heidelberg 1971. [9] BALCERZYK S., Introduction to Homological Algebra (Polish), BM vol. 34, Warszawa 1970. [10] BIALYNICKI-BIRULA A., Linear Algebra with Geometry (Polish), BM vol. 48, Warszawa 1976.

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References

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[21] GUNNING R.C., Lectures on Complex Analytic Varieties, Finite Analytic Mappings, Math. Notes, Princeton Univ. Press, Princeton 1974. [22] GUNNING R.C., ROSSI H., Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs 1965. [23] HARTSHORNE R., Algebraic Geometry, Springer, New York 1977. [24] HERVE M., Several Complex Variables, Local Theory, Oxford Univ. Press, London 1963. [24a] HODGE W.V.D., PEDOE D., Methods of Algebraic Geometry, vol. 2, Cambridge Univ. Press, Cambridge 1952. [25] HORMANDER 1., An Introduction to Complex Analysis in Several Variables, Van Nostrand, Princeton 1966. [25a] KAUP 1., KAUP B., Holomorphic Functions of Several Variables, Walter de Gruyter, Berlin 1983. [25b] KRAFT H., Geometrische Braunschweig 1985.

Methoden

m

der

Invariantentheorie,

[26J KURATOWSKI K., Introduction to Set Theory and Topology, Pergamon Press, Oxford 1961. [27J LOJASIEWICZ S., Ensembles semi-analytiques, LH.E.S., Bures-sur-Yvette 1965. [28J MARTINET J., SingulariUs des fonctions et application8 differentiables, P.U.C., Rio de Janeiro 1974. [29] MILNOR J., Singular Points of Complex Hypersnrfaces, Annals of Math. Studies 61, Princeton Univ. Press, Princeton 1968. [29a] MUMFORD D., Introdnction to Algebraic Geometry, preprint, Harvard Univ. [30] MUMFORD D., Algebraic Geometry I, Complex Projective Varieties, Springer, Heidelberg 1976. [30a] MUSILI C., Postulation formula for Schubert varieties, J. of Indian Math. Soc. 36 (1972) pp. 143-171. [31J NAGATA M., Local Rings, Wiley Interscience, New York 1962. [32J NORTHCOTT D.C., Ideal Theory, Cambridge Univ. Press, Cambridge 1953. [33J NARASIMHAN R., Introduction to the Theory of Analytic Space.'], Lecture Notes in Math. 25, Springer, Heidelberg 1966. [34] RUCKERT W., Zum Eliminationsproblem der Potenzenreihenideale, Math. Ann. 107 (1932), pp. 259-181.

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en, Bull. Ac.

NOTATION INDEX

Chapter A A

;:::j2 E+F,EF,Ez 2

g(M) (minimal number of generators) 4

P(X) 21 edim A 41 rank I 41 v (discrete valuation) 44 An 46,48 n(k) 48, 51

L(X, Y) 86 Lo(X, Y) 86 Gk(X) 87 P(X) 87 Bk(X), B(X) 87 a

= ak = aX = af

T

= Tk = T X = T{

Vi. 89

Sk(U) 90

89

n(V) 91 E-, 5- 93 L.93

In 48

G~(X) 93

h(I) (height of an ideal) 55

P~(X) 93

dim A (dimension of a ring) 56 syzM 57 hd M (homological dimension)

Gk(P) 95

60

87

Chapter C I

dad,

/I

dabf 108

prof M (depth) 63

f!