Topics in Algebraic and Analytic Geometry. (MN-13), Volume 13: Notes From a Course of Phillip Griffiths 9781400869268

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Topics in Algebraic and Analytic Geometry. (MN-13), Volume 13: Notes From a Course of Phillip Griffiths
 9781400869268

Table of contents :
Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
Bibliography

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TOPICS IN ALGEBRAIC AND ANALYTIC GEOMETRY Notes from a course of PHILLIP GRIFFITHS Written and revised by JOHN ADAMS

PRINCETON UNIVERSITY PRESS AND UNIVERSITY

OF TOKYO PRESS

PRINCETON, NEW JERSEY 1974

Copyright (c\

1974 by Princeton University Press

Published by Princeton University Press, Princeton and London All Rights Reserved L.C. Card: 74-2968 I.S.B.N.: 0-691-08151-4

Library of Congress Cataloging in Publication Data will be found on the last printed page of this book

Published in Japan exclusively by University of Tokyo Press in other parts of the world by Princeton University Press

Printed in the United States of America

Introduction This is a revised version of the notes taken from a class taught at Princeton University in 1971-1972.

The table of contents gives a good

description of the material covered. The notes focus on comparison theorems between the algebraic, analytic, and continuous categories.

CONTENTS

Chapter One Section 1.

Sheaf theory, ringed spaces

1

Section 2.

Local structure of analytic and algebraic sets

9

Section 3.

Pn

19

Section 1.

Sheaves of modules

23

Section 2.

Vector bundles

33

Section 3.

Sheaf cohomology and computations on

Chapter Two

P

45

Chapter Three Section 1.

Maximum principle and Schwarz lemma on analytic spaces

56

Section 2.

Siegel's theorem

61

Section 3.

Chow's theorem

69

GAGA

73

Chapter Four Section 1. Chapter Five Section 1.

Line bundles, divisors, and maps to

P

83

Section 2.

Grassmannians and vector bundles

Section 3.

Chern classes and curvature

112

Section 4.

Analytic cocycles

128

94

Chapter Six Section 1.

K-theory and Bott periodicity

136

Section 2.

K-theory as a generalized cohomology theory

144

Section 1.

The Chern character and obstruction theory

154

Section 2.

The Atiyah-Hirzebruch spectral sequence

164

Section 3.

K-theory on algebraic varieties

183

Section 1.

Stein manifold theory

196

Section 2.

Holomorphic vector bundles on polydisks

203

Chapter Seven

Chapter Eight

Chapter Nine Concluding remarks Bibliography

215 217

1

Chapter One §1

Sheaf theory, ringed spaces On

(E , the space of η complex variables, there is a sequence of sheaves

defined in the usual topology: cont

diff

hoi

alg

U c (E

where for an open set

T(U, O ) = continuous complex-valued functions defined on U ' r*rmt cont OO

T(U, O,

)= C

T(U, O ) = T(U, O

)= rational holomorphic functions defined on

U has a neighborhood q

take

W

in

U is said to be rational just in case each U

W , with

(E [z , . . . , ζ ]

U.

such that there are two polynomials φ= p/q

in

a

W

W . In fact, because the

is a unique factorization domain, one can always

is an open set properly contained in the open set

always an element of O, . But diff

O, , hoi

theorem about this.

T(W, O and

in

p,q

W=U.) If

for

φ on

nowhere zero in

polynomial ring

U.

holomorphic functions defined on U .

(A holomorphic function

with

complex-valued functions defined on

) which does not extend to O , alg

behave differently:

U then there is U , and similarly

We will prove a little

1.1.2

2

l.A THEOREM (Hartog's removable singularities theorem) In case W=U

less a point, every holomorphic function on W

We may suppose that such that f

for

Λ (6)= { ζ :

W = U - {(0, . . . , 0)} . If

sup j ζ. | < 6 } i=l, . . . , n

extends to

n>2

and

U.

δ is a small positive number

is contained in

U we define for each

holomorphic in W

ζ

in the interior of the polydisk. It will suffice to show that

f=f

in the

interior of the polydisk less its center. But for a point inside the polydisk with ζ / 0

the formula

Γ

Kz 1 ,...,Zj n'= 2ir/-l

!

«?,z2,...,zj

Iz 1 I= 6 is valid, so

f=f

1

U-

where both are defined.

This type of behavior is more pronounced in the case of holomorphic rational function on W will extend to nomial with zeroes in The sheaf

O

d5

2

U but no zeroes in

O .

. Every

U unless there is a poly­

W .

may naturally be restricted to a coarser topology on (E ,

the Zariski topology . A set in

(E

is a Zariski closed set in case it is the locus

of zeroes of a set of polynomials— one can always take the set of polynomials to be an ideal in the polynomial ring. The Zariski closed set associated to an ideal be denoted

I will

V(I) ("variety of I") . The complement of a Zariski closed set is a

I

3

Zariski open set. This defines a topology on (E , coarser than the usual topology. Two algebraic facts about this topology are (1) V(I)= φ just in case

I=(E[Z , . . . , ζ ]

(2) V(I)= V(J) just in case rad (I)= rad (J)

.

The first of these facts is called the Hubert Nullstellensatz. For a proof of this, and the deduction of (2) from (1), see Safarevic [35] , or Lefschetz [ 20] . Henceforth when we consider the sheaf

O , on (E alg respect to the Zariski topology, sometimes denoted (E iiar

it will usually be with

((E11, O ) , ((En, O ,. „ ) , (·

In case X is an arbitrary algebraic variety it is covered by affine opens, X = U.X..

For each i we get an analytic (X., O, .) and a morphism

ι· V

1®g - g®1

The proof that this is an isomorphism is purely algebraic and we omit it.

See

Mumford [25 ] . A word about the geometric meaning of the sheaf Ω γ/q-, is in order: For a point a on an analytic or algebraic variety x, the vector space (Ω ._) /ma(Q Λ/

(L a

._) Jy (k a

is to be thought of as the dual to the vector space generated by the tangent directions

29 II. 1. 7

to Xat a. We suppose that X is an analytic subvariety of an open set U in C , and that a is the origin.

A complex line through the origin is tangent to X at 0 in case it

is a limit of secants through pairs of points on X as those pairs approach 0. element f of (Ω

An

) /m (Ω„ ) should define an hyperplane in the vector space A/(C a a A/CL a

generated by those tangent lines — w e ' l l give a rough description of how this works. A tangent line is defined by a sequence {(a , a ) } in χ χ χ - Δ approaching (0, 0) with η, and defining a line I in the limit. I is in the hyperplane defined by 2 f e I Jl. . (I. = stalk at (a,a) of diagonal ideal on X xX) just in case (a, a) (a, a) (a, a) iJTT, η -> =°

f

(a„>al) TT T ι

-

η

Ka n , a ) η

η

Now we can investigate the geometric signifigance of the coherence of the sheaf Ω

. First note that, for a coherent sheaf F on an analytic space or algebraic

variety X, dim-, F / m F UJ

cl

el

is the minimal number of elements which generate F 3.

d,

over O — t h i s is a consequence of Nakayama's lemma.

From coherence conditions

one sees thatdim F / m F is an upper semi-continuous function of asX, in fact for any η the set {a: dim F / m F } > n is defined locally by distinguished functions. Consider the case of X an open set in (C .

Here dim (Ω UJ

dim _ m /m

X/

) /(Ω UJ

a

) m Λ/ UJ a

di

- n a t every point.

2 One knows in general that for any local ring of a point O , dim m /m = minimum number of generators of m over O (from Nakayama's lemma again).

30

II. 1. 8

From our earlier discussion of non-singularity we get the II. J. THEOREM Let χ be an analytic space.

The set of points at which X is a

complex manifold is the complement of an analytic subvariety. From the definition of dimension one sees that dim Xis upper semicontinuous in a, and that X is non-singular at a just in case dim χ = dim (Ω If

) /m (Ωγ/_) .

is irreducible then dim Xis constant, say at n, since the regular locus is

connected.

Then the set of singular points is the set

The general case follows from this. Similarly, there is the II. K. THEOREM The set of points at which an algebraic variety is singular is an algebraic subvariety. All this gives a nice interpretation of the behavior of the coherent sheaf Ω 2 2 3 2 Consider for example the analytic subset of (E given by y = χ + χ

X/C

31 II. 1. 9

T h i s will be a complex manifold e v e r y w h e r e except at (0,0). dim m / m for a = (0, 0) w h e r e the n u m b e r i s 2.

2

= 1, except

Note that the tangent s p a c e to the v a r i e t y at (0, 0)

contains two l i n e a r l y independent lines. We'll now give a brief d i s c u s s i o n of an a s p e c t of the t h e o r y of analytic space (and a l g e b r a i c v a r i e t i e s ) , the p r o b l e m of the r e s o l u t i o n of s i n g u l a r i t i e s .

We s t a t e

the p r o b l e m formally: Suppose X i s an analytic space.

Does t h e r e exist a complex manifold X' and a

holomorphic m a p φ·. X' — > X such that (1) φ i s p r o p e r and surjective (2) If y i s t h e s i n g u l a r locus of x, then φ: X - φ

(Y) — > X-Yis an i s o m o r p h i s m of

complex manifolds. T h i s h a s been p r o v e n by Hironaka [ 15 ] . The s i m p l e s t e x a m p l e s of the r e s o l u t i o n of s i n g u l a r i t i e s i s in the c a s e of onedimensional algebraic varieties.

F o r e x a m p l e if χ i s the v a r i e t y in C

2

V(z

2

3 - ζ )

with s i n g u l a r point (0,0) then X = (E

and the m a p C — >

is given by ζ — > (ζ , ζ).

One n e e d s m o r e m a c h i n e r y than we have at our disposal to d e s c r i b e the i n t e r e s t -ing e x a m p l e s of the r e s o l u t i o n of s i n g u l a r i t i e s in h i g h e r d i m e n s i o n s .

We will

32

Π. 1.10

mention one example of a surface: Let X = V (XT + XT + XT ) singularity at the point (1,0,0,0).

> P , with a

This singularity is resolved by an X, a complex

manifold which is topologically P x P

but with a different complex structure.

33

II. 2.1 Chapter Two § 2

Vector Bundles An important class of coherent sheaves on analytic and algebraic varieties arises

from vector bundles.

We first recall the definition: If X is a topological space a

(complex) vector bundle of rank η over X is given by a topological space over X, φ : Y — > X, with the property that there is a covering {U.} of X such that there are isomorphisms φ

-1/,, , ~ ,, „n (U.) - ^ > U. χ C ι

1

T]1

with φ restricting to the natural projection; it is further required that the maps

η. η'1·, u.nu. χ u.nu. x GL(n, (E) IJ

such that

η .τ?. IJ

I J

'

: (χ, ν) — > (χ, ν . .(χ)ν) '

'

IJ

If X is a differentiable manifold one gets the notion of a differentiable vector bundle by requiring all maps to be differentiable, using the natural differentiable structures on (C , GL(n, (C). If X is an analytic space one defines a holomorphic vector bundle by requiring Y to be an analytic space and all maps to be holomorphic (using the natural structures on C , GL(n, (E)). A slight modification is required to define algebraic vector bundles over algebraic varieties: Here one requires Y to be an algebraic variety φ to be al gebraic.

Also

m 1 φ v(U.) ^ - > U. χ (E„ ^ ι ι Zar

n2 with 77. algebraic. GL(n, (C) is a Zariski open of (C and so has a natural algebraic

34

II. 2. 2 structure; one requires the maps ν .. to be algebraic. If Y

φ

> X is a vector bundle (continuous, differentiable , holomorphic or algebraic)

one defines a sheaf of O , 0,.,,, O, ,, or O , - modules Γ (Y) cont' diff hoi' alg Γ(υ,Γ(Υ)) = sections of Φ : rf>"\u) — > U These are sheaves of modules because one has the isomorphisms φ _1 (U.) -^-> U. χ { isomorphism classes of locally free sheaves of rank η on X} We want to show that this is an equivalence, that every locally free sheaf (up to isomorphism) arises in this way and that a vector bundle is determined (up to isomorphism) by its locally free sheaf. We first take another look at the data which describes a vector bundle. Among other things we get an open covering {U.} and maps (continuous, differentiable, holomorphic, or algebraic). V .. : U.n U. — > GL(n,(C).

35

II. 2. 3

Because of the relation between

and

we know that

Now suppose we are given a covering and maps satisfying

Then we can con struct a vector

bundle by pasting together

along the sets

in the obvious way; the condition on triple overlaps allows us to do this consistently. We call the set of all such maps

with respect to the covering

where O might be

We have shown that to every vector

bundle which is defined according to the covering

we can associate an element of

and conversely

A morphism of vector bundles

is given by a map

such that

commutes, and with the property that there is a covering both Y and Y' are defined and such that the map induced by the trivializations on isNote of the form where that the maps

of X according to which must satisfy

36

II. 2. 4 (All maps will be required to be differentiable, continuous, holomorphic, or algebraic, according to context.) Conversely, from a collection of maps {φ .} satisfying ν

φ - φ ν' J i ij

ij

we can construct a morphism of vector bundles. From this we see that a necessary and sufficient condition for two elements {i>..} and {v'..} of Z to define isomorphic vector bundles is the existence of maps ψ . : Xi.

> GL(n, C) such that ν . φ. = φ .ν' ij

J

ι

IJ

We say that two elements of Z ({U.}, GL(n,0)) are equivalent just in case such maps exist; the quotient by this relation is called H ({U.}, GL(n,0)). Suppose that {W.}-u is a refinement of {U.}, so that for each -t there is ρ (-£• ) such i that W, c U . . . .

Then there is defined a map

H \ { U . } , GL(n,0))

> H1C(W4I, GL(n, O))

The direct limit of these sets over all coverings is denoted H (X. GL(n, O)). It is equivalent to the set of isomorphism classes of (continuous, differentiable, holomorphic, or algebraic) vector bundles on X. Now we will show how, given a locally free sheaf of constant rank, to associate an element of M (X,GL(n, O)) to it. If L is that sheaf (of rank n) pick an open cover {U.} so that there are isomorphisms of O-modules restricted to U. Ll

U. ι

^->0 X one knows from local considerations that T(Y) is

locally a direct summand of f*T(X); then there is a quotient bundle on Y, f* TXX)/T(Y) = Ν the normal bundle of Y in X.

χ / γ

The rank of the normal bundle will be dim X -dim Y

= co dim Y. To appreciate the geometric significance of the normal bundle , note that if M

> Y is a holomorphic vector bundle (of rank n, say) then M naturally contains

Y as a closed submanifold : Y is imbedded as the zero section of M. And the normal bundle of Y in M is j ust M. It is a theorem in differential topology (see Milnor [23 ] )

that in case Y — > X

has normal bundle M then there is a neighborhood of Y in X which is differentiably

39 II. 2. 7 isomorphic to a neighborhood of Y in M. But it will be in general impossible to pick holomorphically equivalent neighborhoods.

In any case, the normal bundle of

an imbedding gives some information about a neighborhood of Y in X. On a non-singular algebraic variety X the sheaf Ω

will be locally free,

thus defining a vector bundle on that variety, the bundle of differentials. The dual of this bundle is called the tangent bundle on a non-singular algebraic vareity; to justify this definition we observe that on a complex manifold X

Ω .

is naturally isomorphic

to the sheaf of sections of the dual of the holomorphic tangent bundle. To see this, note that the normal bundle of X Y

> XxX is just T(X). Now if

> X is a closed imbedding of complex manifolds with I the sheaf of ideals

2 defined by Y then I/I is a coherent sheaf on Y. modules

„ Horn (I/I ,O y )

> Ν

The idea is to get a map T i

Λ I Y

There is an isomorphism of O -

χ / γ

2 > Horn (I/I , O ) by differentiating a function I

in I along a tangent direction. Of course if the direction is tangent to Y the derivative will be O, so the kernel of this map is just T . Applying this to the diagonal ideal brings us back to our previous viewpoint on Ω

χ/α· An important special class of vector bundles is the class of line bundles, or vector

bundles of rank one. The set classifying them is HVjGL(I 1 O)) = H \ X , 0 X ) Two locally free sheaves of rank one L., L are multiplied by

(W-

> L

lS

L

2

40

Π. 2. 8 This is also a locally free sheaf of rank one, so we have defined a multiplication of line bundles. The inverse of a locally free sheaf of rank one, L , is Hom_ JX„0) because Horn (L ,0) ® L. -^—> O. Because of the existence of inverse, locally free sheaves of rank one are called invertible sheaves. One geometric way in which line bundles arise is in the consideration of Cartier divisors: an effective Cartier divisor on an analytic or algebraic variety X is given by a covering {U.} of X and for each i an f. εΓ(ϋ.,Ο) in any stalk of U., and so that V .. f. = f. '

l'

IJ J

such that f. is not a divisor of zero

on U.n U., V .. a unit in Γ (U.fl U.,O). Two ι

1

j'

IJ

ι

J'

effective Cartier divisors |(U.,f.)j and{W.,g.)j are the same in case {W-divisors}

which is an injection, and in fact exhibits the group of W-divisors as generated by the image of this map. In the case of an algebraic variety with singularities one may not even be able to define the map properly. One difficulty is that not necessarily every subvariety of codimension one is given locally by a single equation. For example, the point 2 3 2 a = (0,0) on the variety V(y -(x +x )) pictured before is a subvariety of codimension

42

II. 2.10 one, but if the ideal of functions vanishing at this point could be gens rated by one element then one would also have

while we have seen that this

dimension is two. We'll give some more concrete examples of line bundles now.

The most

important line bundles in algebraic geometry are certain line bundles on projective spaces, both algebraic and holomorphic. in

and

Pick homogeneous coordinates

Let p denote the point

and

define a map from

This map (called the projection from p) is both an algebraic and holomorphic map. The preimage of a point

is the set of all points

in other words, a line. In fact,

and the map to

corresponds to projection on the first n terms. is naturally isomorphic to

Similarly, for all i,

On

to these natural maps is

the transition function associated

The line bundle thus defined is called the

tautological line bundle on projective space - either algebraically or holomorphically. Its sheaf of sections is denoted either

or

As a matter of notation,

the dual of this sheaf will be denoted of

or its dual will be denoted for all m,

the m-th tensor power or

provided we understand

There is an alternative description of the sheaves and holomorphic cases.

so the structure sheaf . in both the algebraic

We have the maps

which define the structures on ;

For

we define a sheaf of

-modules F(m)-

43

II. 2.11 either algebraically or holomorphically - by {algebraic or holomorphic functions f such that for all Now the sheaf F(m) is locally free of rank one: On

the map

reduces to

Given U open in

, f algebraic or holomorphic on

define

then

we define

by This defines an isomorphism of sheaves on so this sheaf is invertible.

The same thing happens over

The transition functions

F(m) with respect to the covering

associated with so that

All this is either algebraic or holomorphic. There is a natural homomorphism

of C-vector spaces, the definition of which is obvious from the last definition of these

sheaves.

We'll prove a little theorem comparing the analytic and algebraic

situation III. X

THEOREM The map

44

II. 2.12 is an isomorphism.

There is a natural isomorphism {homogeneous polynomials of degree m in

An element f of

is a holomorphic function on for all z.

shows already that

We must have series representations. m.

By Hartogs' theorem f is holomorphic in

unless m

such that (This

Represent f as a power series

by the uniqueness of power This shows that f is a homogeneous polynomial of degree

45

Π. 3.1

Chapter Two § 3

Sheaf cohomology and computations on P .

Much of our work in these notes will use the tool of sheaf cohomology theory, which we will now recall.

Details are in Godement [ 9 ], and Swan [ 34].

If A is a sheaf of abelian groups on a topological spaceX, there are defined, for all i > 0, the cohomology groups Η (X, A). They can be introduced as follows: A sheaf of abelian groups is called flabby if a section of the sheaf over any open set can be extended to a section over the full space.

For any sheaf of abelian groups A on X there

is an exact sequence extending indefinitely to the right 0

> A

where the F. are flabby.

> F„ 0

> F, 1

> ...

> F

>... η

We set

HX(X,A) = ker F„ 2

>Γ(Χ, F . + 1 ) ) / l m(T (X, F. ^)

i > 0, and H°(X, A) = ker(T(X, F°)

> T(X 5 F.))

> Γ (X, F ) )

It can be shown that these groups do not depend on the particular flabby resolution of A.

In case A is a module over some sheaf of rings 0, then one can choose the F. to

be 0-modules, so that the groups H(X, A) are Γ (X, 0)-modules. Some properties of the cohomology groups are: There is always a natural isomorphism H (X, A) ——>1P(X, A). Any morphism of sheaves A H^X, A)

> B induces a morphism of cohomology groups

> H\X, B) for all i.

Given an exact sequence 0

> A

> B

> C

> 0 of sheaves of abelian

46

II. 3. 2

groups there are defined, for all i > 0, maps δ.: H (X, C) — > H

(X, A), so that

the long sequence 0 — > H°(X, A) — > H°(X, B) — > H°(X, C) -§>-> H1 (X, A) — > . . . is exact.

In case the sheaves are sheaves of modules over a sheaf of rings, all the

maps are Γ (X, 0)-homomorphisms. A consequence of this is the following:

Suppose

0 — > A —> B —> B —> B —> . . . is an exact sequence of sheaves of abelian groups, such that H (X, B.) = 0 for all i > 0, all j .

Then

H*(X,A) -^->Ker(H°(X,B.) — > H°(X, q + 1 ))/ilm(H 0 (X,B M )

H°(X,B.))

This makes it important to find cohomologically trivial sheaves, that is, sheaves F such that H (X, F) = 0 for i > 0.

Flabby sheaves are cohomologically trivial.

Other important examples of such cohomologically trivial sheaves are the soft sheaves:

A sheaf of abelian groups A on a Hansdorff space is called soft if to any

covering {U.} of X there is a family {φ.} and only a finite number of the maps Q . at any stalk.

of endomorphisms of

A such that

φ.=0 off U.

are non-zero at any stalk, and Σ φ.

= id

A partition of unity argument shows that the sheaf 0 . , on any differ

-entiable manifold is soft; this will be the most important soft sheaf for us. Closely related to the cohomology groups of sheaves are the Cech cohomology groups: First, from an open cover {U.} of X we define the groups H ({U.}, A) as follows: Define C lf ({U.}, A) as the set of all maps f which to each ι+Ι-tuple

47

II. 3. 3

of opens in

assigns an element

in

such a way that the association is alternating - that is, and structure.

differ by a transposition.

if has a natural group

There is a group homomorphism

given by

and the group

is defined as Ker

(We remark that one could

drop the alternating requirements and work with the groups thing.

The resulting groups

Given a refinement

doing the same

would be isomorphic). so that

we can define maps

By taking the direct limit over all coverings, one gets the Cech cohomology groups

It is not always the case that

in fact the cohomology groups

fail to have the nice exact sequence properties of the groups

The

relation between these two cohomology theories is as follows: For any covering of X there is a spectral sequence with

term

converging to

From consideration of these sequences one deduces (1) For all i there are maps For i = 0 or 1 this is always an isomorphism.

leading to maps

48

II. 3. 4

(2) Suppose {U.} is a covering such that H (U.,Π . . . Π U. ,A) = 0 for i > 0 and any L, . . . , j

in theindex set.

Then H1C(U.]-, A) -=^-> H^X, A) (Theorem of Leray). For more discussion of this important point, see Godement [ 9 ] . There is a similarity between the )£ech cohomology groups and the sets ΗΎΧ , GL(n, O)) which classify vector bundles. We can interpret these sets as 1

1

χ

cohomology sets of sheaves of non-abelian groups — and H (X, GL(I1O)) = H(X 1 O 1 , the first cohomology of the sheaf of units. The group structure on H (X, 0 ) is the same as that otherwise defined on the line bundles. It is by reference to the theorem of Leray that one can, in some cases, compute sheaf cohomology groups.

What is needed is a class of cohomologically trivial open

sets to make a covering.

These are known to exist, in the case of coherent analytic

sheaves, by the ILL

THEOREM (Cartan's Theorem B) Suppose that F is a coherent analytic sheaf on the analytic space X, where X is a

closed analytic subvariety of a polydisk in C . Then H (X, F) = 0 for i > 0. This theorem has its algebraic analogue. II. M THEOREM (Cartan's Theorem B-algebraic version) Suppose that F is a coherent algebraic sheaf on the algebraic variety X, where X is a closed subvariety of c " .

Then Η \ Χ , F) = 0for_i > 0.

49 II. 3. 5

Since any analytic space or algebraic variety has a covering by such things, these theorems - together with the appropriate facts about intersections - tell us that cohomology can be computed by the Cech method. Gunning-Rossi [13]

and Serre f27] .

These theorems are proved in

We shall discuss them more fully in

Chapter Eight. It is also sometimes possible to compute the cohomology of coherent analytic sheaves through the use of soft resolutions.

This technique works when X is a complex manifold.

On a complex manifold M there is the sheaf

Ω,

of differentiable,

complex-valued

one-forms, which is a sheaf of 0, ,-modules (this is not a coherent sheaf). ' hoi splits into a direct sum of 0

-modules, Ω ,.„ = Ω ' ®Ω ' :

This sheaf

If U is an open polydisk

in C then Ω'

= subsheaf generated by dz , . . . , dz over 0

Ω ' = subsheaf generated by d z " , . . . , dz Any holomorphic automorphism of U takes Ω '

into itself and Ω '

into itself.

This

shows that there is such a decomposition on a complex manifold. We get a corresponding splitting of s-forms

Ω,

=Σ_

Ω '

where

The operator of exterior differentiation splits into d = S + 3 , where 3 : ΩΡ, n P ' q

, ?: n P ' q — > Ω Ρ ' q .

Again, this is clear on a polydisk, where

it is invariant under holomorphic automorphism.

50

II. 3. 6

An important local theorem is II. 0 THEOREM (Poincarfe lemma) The sequence of sheaves

is exact, if n = dim M. As a consequence of this we get de Rham's theorem:

We want to investigate the similar situation for the

-operator.

The

-cohomology

groups of the complex manifold M are defined by

If N is a complex manifold, a holomorphic map

induces maps

This is functorial. These maps depend only on the holomorphic homotopy class of f: Letting suppose we have a positive degree. Then we get forms on M by We will show that -exact. Letting w be a local holomorphic coordinate on M, we can locally write

where a (z, w) involves no

Set

Then

implies that

and

51

II. 3. 7

Since

we get what we wanted.

Now if

is a holomorphic map of complex manifolds, this shows that

the maps on the same.

-cohomology

b

y

a

r

e

This implies the

II. P THEOREM If w is a

induced

-Poincarfe lemma)

-closed form of positive degree on the polydisk U, then w is

-exact on

any smaller polydisk in U. Let W be a smaller polydisk contained in U, so zW

some small

The map

given by to a point.

exhibits a holomorphic homotopy between the injection and the map This proves the theorem.

We can also formulate this theorem as: The sequence of sheaves is exact. V

Now we'll compute some cohomology groups by the Cech method. THEOREM II. Q

52

II. 3. 8

The part about the

has already been done.

the fact that

for

The proof of the rest will use which will be discussed in

Chapter Eijfit. We shall also use the fact that any holomorphic function on can be represented by a Laurent series in several variables,

This can be proven, as in the 1-variable case, from Cauchy's integral formula. (Note that the existence of Laurent series representations on

Fix homogeneous coordinates

and identity

with affine coordinates notation

is equivalent to

with

We shall use the f o r e t c .

A holomorphic function f on

can be

n factors written

; the condition that f be the restriction

of a holomorphic function defined on

is that

monomials in the power series, while only Denote by N the covering

Now if

an element of

for all non-zero

can be negative. By the theorem of Leray,

is given by a family of sections

53

II. 3. 9

satisfying the cocycle rule. data can be expressed as f u n c t i o n s h o l o m o r p h i c in

This

(use

satisfying for

with the

restriction that

extends to

(What we've done is systematically

trivialize a section

(d)) as a function over a subset of

We want to show that any such thing is a coboundary, proved that to find, for any

It is plain that a choice of

holomorphic in

for every

with the equations (*), determine all the other arbitrarily and get such a solution, defining

We want such that

would together . We could choose

and the formal relations would be satisfied. But there is nothing to guarantee that comes by restriction from a function defined in be checked is that in the Laurent series for

What must

54

II. 3.10

there are no monomials of weight greater than d, so we must choose so as to get rid of such terms in

etc.,

Now if q Δ (r) for some r and η which is holomorphic and finite. is, there is an analytic subset D of Δ (r) such that tr: U — π

That

(D) — > Δ (r) — D is a

map of constant rank and a topological cover. The degree of the cover is called the degree of the map. We can assume that ν

({0}) ={x}.

For a proof, see Narasimhan [ 26 ] . Now we'll give some applications of this theorem. III. B. PROPOSITION (Maximum principle in several variables) Let f: Δ (r) — > (E be a holomorphic function. J f | f | is a local maximum at 0 then f is constant. Given zeC n , define f (u) for small ueC by f (u) = f(uz). By the one variable maximum principle, f is constant. The proposition follows from this. Resolution of singularities easily gives an extension of this to analytic spaces. We can also do this by branched coverings. III. C. PROPOSITION (Maximum principle for analytic spaces) Let f: X — > C be a holomorphic function on an analytic space, with |f| a local

57

III . 1. 2 maximum at xeX.

Then f is constant in a neighborhood of X.

We may assume that X is irreducible and that there is a d-sheeted analytic cover.

where

For

with we set

We may assume that f is a maximum at Since

, and

is bounded it can be extended to a holomorphic function on

(see Gunning-Rossi [

Suppose that f is not constant.

13 ], Chapter I).

Now

Then it follows from the maximum principle on

complex manifolds, using the irreducibility of X, that from this we see that

when

while it must be that

and

Since D has measure 0, this implies identically.

The Schwarz lemma also admits generalization. II. C.

PROPOSITION (Schwarz lemma in several variables)

Let f be a holomorphic function on

with

< M everywhere and with a

zero of order h at the origin. Then Fix and lei

everywhere, as one sees by letting applying the maximum principle.

Now

and

In particular,

The Schwarz lemma for analytic spaces is stated in terms of branched coverings. II. E. Let

PROPOSITION (Schwarz lemma for analytic spaces) be an analytic space exhibited as a d-sheeted branched

58

III. 1.3

covering of the polydisk.

Let f be a holomorphic function on X with

suppose that

M and

Then

for all xeX.

Once again set

on

and extend. Even at a point

although some

may be repeated.

we have

Applying the Schwarz lemma to

we get

The proposition now follows from the III. F.

LEMMA Let

be complex numbers.

Then

We omit the proof of this. As a final example of the use of the technique of branched coverings we shall the III. has generalize sequence G. We limit a holomorphic PROPOSITION may isofthe holomorphic assume holomorphic theorem limit. thatabout the at functions the origin the origin. completeness is converging beaapoint Perhaps hypersurface ofuniformly of V restricting the and space itinwill on theto of compact suffice polydisk aholomorphic slightly to subsets show smaller functions. that ofA_ V

59

III . 1.4

polydisk, we may assume that f is a distinguished pseudo-polynomial

The projection to the z' hyperplane exhibits V as a d-sheated branched covering which is ramified over those points where the discriminant of the polynomial vanishes - call this D. Given g analytic on V define

for points in the

polydisk with zYD by

After perhaps restricting to a slightly smaller polydisk there will be a constant M - independent of g - such that

This implies that o

be extended to the entire polydisk, since we may assume g bounded.

can

The function

will satisfy

for Now given a convergent sequence associate to it a sequence relation

we can in a neighborhood of the origin

also convergent, with holomorphic limit

p g, with

.

The

shows that g is holomorphic.

Two remarks about this proof are in order: First, we have generalized the completeness theorem to hypersurfaces.

The technique of branched coverings,

properly applied, will extend the theorem to other codimensions - see Narasimhan Second, the same proof works under the weaker hypothesis that the

60

III.l. 5.

g are defined, holomorphic, and bounded only on the regular points of V. The sheaf of such functions is called the sheaf of weakly holomorphic on V and appears in the process of normalization.

For more details, see Narasimhan [ 2 6



61 III. 2.1 Chapter Three § 2

Siegel's Theorem Now we'll put some of these results to work and prove our first comparison

theorem, a theorem about meromorphic functions.

We first give an algebraic

description of these objects: If X is an analytic space on algebraic variety one forms a presheaf K

by T(U, K χ ) = total quotient ring of T(U,0 )

The associated presheaf is denoted by K . In case X is an irreducible analytic space Λ

or algebraic variety, Γ (X, K ) will be a field, the field of meromorphic or rational Λ

functions on X. A meromorphic or rational function is thus defined by a covering {U.} and functions ω. φ. = φ.φ. οηϋ.Π U.. If X is an irreducible algebraic variety then K is a sheaf of fields and in fact a X constant sheaf. We denote this field by K

(X). It is a finitely generated extension

of (C with transcendence degree equal to the dimension of X. For details on this, see Lefschetz [

20

] 0 r Safarevic [

35

].

Also if X is an irreducible algebraic variety the natural analytic structure on X (in which X is irreducible, as will be proven in the next section) gives rise to the field of meromorphic functions K (X) and there is an injection of fields K (X) — > K (X). J ^ mer rat ' mer v ' We want to show that, in case X is a complex manifold, meromorphic functions on X can be interpreted as quotients of sections of line bundles. First, given a line bundle with transition functions JCK..} and two sections {φ .}, {

The projection ff of X onto the first n-1 coordinates

is a finite analytic map with branch locus B^ (C

. For ζ V B there are locally defined

holomorphic functions χ , ( ζ ' ) , . . . , x . ( z ' ) which are permuted among themselves by analytic continuation throughout C

-B, the x.(z') being the roots of

ζ + 1ρ , ,(z')z +. . .+p_(z'). r η Cl-I η 0v

If ff ((E

-B) is not connected then we can

70

III. 3. 2. divide

in to two groups

which are nonempty and never confused by analytic continuation.

Setting

elementary symmetric function of

elementary symmetric function of

the

and

are holomorphic and single-valued on

and extend

across B. There is a factorization

and we would have a contradiction if all the The following argument shows that the therefore polynomials. and consider

Consider

imbedded as

could be shown to be polynomials. s have polynomial growth and are

with homogeneous coordinates with

The strong topology closure of

the hypersurface

does not contain the point with homogeneous coordinates implies the existence of for all

such that This gives a bound

which

71 III. 3. 3

(l+|p d _ 1 (z'1j+. . . + | p 0 ( z ' ) h > e Ix1(Z") with e independent of i, and this is what we wanted. A consequence of this is that any Zariski open subset of an irreducible affine algebraic hypersurface is itself analytically irreducible.

It is a fact from

algebraic geometry that any irreducible algebraic variety contains a Zariski-dense Zariski open which is equivalent to an open on an irreducible hypersurface (because one can find a hypersurface with the same field of rational functions; see Lefschetz [20]

V or Safarevic [35]

for a proof) .

The general case of the lemma

follows from this observation. All we need show now is that dim X = dim V.

The global sections of Omiv(l\ , P l N hoi

must give local coordinates at some point of V so we know that dim H (V, 0,/d)'i grows like a polynomial in d of degree m = dim V.

In fact this argument shows

that

dim(H°(PN, OpN(d)W H V . I ^ ) grows like a polynomial of degree m in d.

Since

Η ° ( Ρ Ν , Ι ^ Μ = H°(P N ,I (d)) we fine that m = dim X. Chow's theorem should be interpreted as a statement that analytic subsets of N C which satisfy a certain growth condition must be algebraic. The growth condition in question is that the closure of each irreducible component in P

N

again

be analytic and irreducible. There are some more comparison theorems which can be deduced from Chow's theorem. We'll give one:

72

III. 3. 4 III. L LEMMA Let φ : X —>X' be an algebraic map between algebraic varieties which is analytically an isomorphism.

Then φ is algebraically an isomorphism.

We may assume that X (and therefore X') is irreducible.

Since every point in X'

has exactly one point of X mapped to it we know that φ is birational; this means that there is a proper algebraic subset D of X' on the complement of which φ inverts algebraically (see Lefschetz

[20]

v

or Safarevic

discussion of birational maps). Then φ

[35]

for a

is algebraic. J X-D

By locally exhibiting X' as a branched algebraic cover of some C and using symmetric functions we are reduced to proving that if f: C — > (E is a bolomorphic map the restriction of which to some non-empty Zariski open is algebraic then f is algebraic.

But this is clear.

III. M COROLLARY Every holomorphic map between projective varieties is algebraic. If φ : X — > Y is the map then the graph of φ is an algebraic subvariety of V

XxY (because the product of projective varieties is projective; see Safarevic [35] ). The projection graph (φ) — > X is algebraic with a holomorphic inverse. is therefore an algebraic isomorphism and this proves the corollary.

It

73 IV. 1.1 Chapter Four

GAGA

§ One In this chapter we shall prove comparison theorems about sheaves and sheaf cohomology.

The groundwork for this has already been laid in the computation

of the holomorphic and algebraic cohomology of the sheaves O n (d) on projective space. If X is a projective algebraic variety we shall sometimes use the notations X alg, for X as a ringed s i - space in the Zariski topology r t=j with sheaf O , a

for X as an analytic space.

l

g

and X, , h o l

There is a morphism of ringed spaces

^ ' "nol

alg

We know from sheaf theory that any coherent algebraic sheaf F will pull back to a coherent analytic sheaf φ * F = F afX

is O , , 8 a,hol

. on X,

.

The stalk of this sheaf at a point

F . The coherence of F, , is a consequence of the rightM & a hol a, alg

exactness of the tensor product. For any i there are natural maps H (X . , F) — > H (X, ,, F

). In general

for a morphism of ringed spaces f:X — > Y and a sheaf F on Y there are induced functorial maps IT(Y, F) — > IT(X 1 ^F). Maps H^Y.F) — > rf(X,f*F) are obviously defiiEd and it suffices to consider these in the cases of our concern, since we can v compute sheaf cohomology by the Cech method off an affine cover. There are nice functorial properties for the association F — > F

·

An O . linear map between coherent algebraic sheaves F — > G induces an O, , linear map F, , — > G, .. hol hol hol Recall that there is a sheaf Horn

(F, G) the global sections of which are the "alg

74

IV. 1. 2

O , -module homomorphisms from F to G. There is a natural map of sheaves alg

which is an isomorphism.

To see this recall that, for coherent sheaves F, G

at a point aeX

and similarly in the holomorphic case.

Then at a point aeX the left side of (*)

is while the right side is

and the map is the natural one. We have reduced the question of the isomorphism of (*) - which it suffices to check at every point - to a question of pure algebra. The isomorphism can be deduced from the following algebraic facts (1) For any exact sequence of coherent algebraic sheaves the sequence

of coherent analytic sheaves is exact. (2) If F is a non-zero coherent algebraic sheaf then ! These again reduce to questions at each stalk. is that for each

Algebraically the point of all this

is a faithfully flat extension of

- the stalk

75

IV. 1. 3

equivalents of (1), (2) serve as a definition of faithful flatness: (1) For any exact sequence

of

modules the sequence

is exact. (2) If A is a non-zero One proves that

module then if faithfully flat over

is non-zero. by noting that both

are noetherian local rings with the same completions. Since our point of view is analytic we shall not go into this, but only use these facts.

For a discussion of the algebra involved see Altman-Kleiman [ l ] ,

or Matsumura [ 22 ] . theorems, more THEOREM is canaThe be unique the stated key comparison natural after coherent to B Ain _If GAGA Ifathe F maps Fis single is article theorems algebraic ais a coherent coherent in is theorem: the generated of which their sheaf two analytic analytic theorems first appear by its appearance, sheaf sheaf in global this oon nsuch achapter sections projective Serre that are there are isomorphisms for [28] iscalled variety a. d^ such the X— The ^GAGA Furtherjthat for that there result allis,i.

76

IV. 1. 4

for each aeP there are {f } e H (P

.,F(d)) which generate F (d) as an

O^i module. hoi, a THEOREM C If F is a coherent analytic sheaf on P, , there is a d_ such that — hoi O H 1 CPj 0 1 , F(d)) = O for i > O, d > dQ. These two theorems are versions of Cartan's theorems A and B respectively, with growth conditions. Cartan's theorem A says that a coherent analytic sheaf on (C is generated by its global sections and the first theorem says that in case this coherent sheaf extends across the hyperplane at infinity the generators can be chosen to have inessential singularities at infinity.

The second theorem bears

a similar relation to Cartan's theorem B. This is not to say that the proofs are directly derived from Cartan's theorems A and B. In fact the derivation of these two theorems will be fairly formal.

The only analysis we'll use is a result about

the finiteness of a cohomology group - although we need Cartan' s theorem B to compute cohomology. First note that it is sufficeint to prove GAGA for X = P . For given an arbitrary projective algebraic

Y — > P , we also have Y,

—> P

., and a

sheaf of O

(resp. O ) - modules is coherent if and only if it is coherent alg hoi as a sheaf of O η (resp. O n ) - modules. And the association F — > F alg hoi gives the same 0 _ n - module whether we consider F as an O or O n nol alg alg module.

Also the computation of cohomology groups can ignore this ambiguity.

Now we'll show how to get GAGA for P out of theorems B and C. PROPOSITION D Suppose that any coherent analytic sheaf F o n P

is a special

77

IV. 1. 5

cokernel - that is, there is an exact sequence of coherent analytic sheaves

such that

are of the form

First of all, to get

so that

the morphism of algebraic sheaves with and take The problem is to show that the map

Then GAGA follows. the natural thing is to consider the cokernel. is algebraic.

Now

so that

and we know that this is the same as its algebraic counterpart.

We can then let

be the cokernel of

We know that

is exact and this ensures that the map

As for cohomology we'll show that

78

IV. 1. 6

is an isomorphism by descending induction on i.

The following lemma starts

the induction. LEMMA E _If_F is a coherent analytic or algebraic sheaf on

then

for i > n. This is because we can compute the group with alternating cochains off a cover with n+1 open sets. We have the exact sequence

where

and the long exact cohomology sequence

and for each i maps

with isomorphism. exact so we (4), the sequence can (5) horizontal conclude are isomorphisms Thissequences that implies (1) is that exact surjective. by inductive (3)and is surjective. everything The assumption five commuting. lemma Weand getwe then a similar know shows that result that (2) (3) isfor an is the an

79

IV. 1. 7

isomorphism. To show that a holomorphic sheaf is induced by a unique algebraic sheaf, suppose that F, G are coherent algebraic sheaves and

Then there is inducing this isomorphism, and

such that

id in

But then it must be that

id algebraically.

The proof of GAGA is thus reduced to showing that every coherent analytic sheaf is a special cokernel.

Such information is provided by theorem B, according

to which there is, for any F a coherent analytic sheaf and a d such that there is a surjective map

which is a start.

We do the same thing to the kernel of this map to get what we

want. Everything is now reduced to the proof of theorem B. We first show that theorem C is a consequence of theorem B. The proof will be by descending induction on the order of the cohomology group, and we can again start the induction because

Represent

F as a special cokernel Giving rise to the exact sequence

There is

such that for

exact sequence

and the

80

IV. 1. 8

s hows that

We'll need this in our proof of theorem B, which is by induction on the dimension n of the projective space.

We start with

where there is nothing

to prove. LEMMA F For a coherent analytic sheaf F on

there is, for each

a

dp such that the stalk F^(d) is generated by the global sections

Pick a hyperplane isomorphic to

of

pa ssing through a.

for all

The ideal sheaf of this is

and we fix a map

For all d

there is the exact sequence which gives the cohomology exact sequence

By inductive hypothesis there is the map

such that

F(d)) is surjective for

If we let d grow we get a long sequence.

with the maps surjective at each stage.

so that

81 IV. 1. 9

THEOREM F The cohomology groups of a c o h e r e n t analytic sheaf on a c o m p a c t analytic space a r e finite dimensional over (C. T h i s t h e o r e m i s p r o v e d in Gunning-Rossi [ is some d

> d such that the m a p s

for

so

d > d

13

]. It i m p l i e s that t h e r e

IL(P , F(d)) — > H (P , F(d+1)) a r e i s o m o r p h i s m s

0—>H°(Pn,F(d-l))—>H°(pn,F(d ))—>H°(Pn"1,F(d)|pn-l) —>0 i s exact for g e n e r a t e for O

d > d .

Pick d

d > d„.

> d

Then the e l e m e n t s of H ( P , F(d)) g e n e r a t e F(d)

- modulo t h e ideal defining P

a

so that the global s e c t i o n s of H ( P

P

a t a.

, F(d) Ι^,η-ΙΊ over

It follows from N a k a y a m a ' s l e m m a

O n

thaf the global s e c t i o n s

H ( Ρ Π , F(d)) g e n e r a t e F(d) . for d > d

T h e o r e m B follows by a c o m p a c t n e s s a r g u m e n t from this l e m m a . COROLLARY

G (Chow's t h e o r e m ) Every analytic subvariety of a p r o j e c t i v e

variety is algebraic. If X i s an analytic subvariety of the p r o j e c t i v e a l g e b r a i c v a r i e t y V then it is the support of a c o h e r e n t analytic sheaf and thus t h e support of a c o h e r e n t a l g e b r a i c sheaf. COROLLARY H Every holomorphic v e c t o r bundle on a p r o j e c t i v e v a r i e t y i s induced by a unique a l g e b r a i c v e c t o r bundle. It m u s t be shown that if F is a c o h e r e n t a l g e b r a i c sheaf and F F i s locally free.

i s locally free then

T h i s r e d u c e s to a local s t a t e m e n t and follows from flatness.

F o r o u r l a t e r p u r p o s e s t h i s l a s t c o r o l l a r y i s the m o s t i m p o r t a n t of the GAGA results.

It s a y s that a holomorphic v e c t o r bundle on an affine v a r i e t y which

s a t i s f i e s a growth condition, to the effect of extending a c r o s s the section at infinity of the v a r i e t y , m u s t be a l g e b r a i c .

We m e n t i o n a n o t h e r i n t e r e s t i n g c o r o l l a r y ,

the

82

IV. 1.10

proof of which is contained in previous remarks. COROLLARY I Every holomorphic line bundle on a projective variety is the line bundle of a divisor.

83 V. 1.1

CHAPTER FIVE § 1

Line Bundles, D i v i s o r s , and Maps to P Every complex manifold h a s a n a t u r a l o r i e n t a t i o n , so that on a compact complex

manifold

M of dimension η t h e r e i s defined a p r e f e r e d g e n e r a t o r of H

(M, 2).

If D i s a complex submanifold of codimension one of the compact complex manifold M then the i m a g e of t h e p r e f e r e d g e n e r a t o r of H H

(M, Z)

and by P o i n c a r e duality a c l a s s

(D, Z) defines a c l a s s

2 [D] in H (M, Z).

[D] in

T h i s i s t h e cohomology

c l a s s of the d i v i s o r . If M i s a possibly n o n - c o m p a c t complex manifold and D i s a divisor, with s i n g u l a r i t i e s , we can still define section.

2 [D] e H (M, 2) , a s we shall s e e

possibly

l a t e r in t h i s

In t h e l a s t section of this c h a p t e r we shall show how to define t h e cohomology

c l a s s of any analytic subvariety of a complex manifold.

Our study of the r e s u l t i n g

analytic cohomology c l a s s e s , o r analytic cocycles,will l e a n heavily on the theory of v e c t o r bundles.

In t h i s section we shall d i s c u s s t h e s e i d e a s in t h e s p e c i a l c a s e of

d i v i s o r s and line bundles. Let M be a complex manifold of d i m e n s i o n n, T h e r e i s a holomorphic line bundle to D.

D an effective d i v i s o r on M.

L — > M, with a h o l o m o r p h i c section c o r r e s p o n d i n g

T h e r e is an exact sequence of s h e a v e s on

M

_ _ exp2ir r -l _x ο — > Z — > O —*> O —> 1 where

2

denotes the sheaf a s s o c i a t e d to the constant p r e s h e a f with stalk

i s induced a coboundary m a p δ: H (M, O*) — > H 2 ( M , Z).

Z .

There

84

V. 1. 2.

Since M is a manifold the second cohomology of M with coefficients in 2 is the same as the second singular cohomology group of M with

coefficients.

kernel of the map 6 is the image of

For a holomorphic line

bundle

is called the first Chern class

The

of L and denoted

One can do the same thing with arbitrary differentiable complex line bundles on M:

assigns to each differentiable complex line bundle its first Chern class.

Hie diagram shows that the first Chern class of a holomorphic

bundle depends only on its differentiable structure.

Furthermore,

because these sheaves are flabby.

so that

Then

on a complex manifold may be identified with the group of complex

differentiable line bundles. THEOREM A Let D be a smooth divisor on the smooth, projective variety M, with holomorphic line bundle

The cohomology class of D agrees with

the first Chern class of L. The proof of this theorem will require ideas to be developed in the rest of this section.

The theorem is actually true in the generality of any divisor on a complex manifold,

although we shall not prove that here.

85 V. 1. 3. The proof will depend on the introduction of a classifying space for complex line bundles, which we shall discuss now. The most important line bundle in algebraic geometry is the line bundle O n (l) on projective space. P

The divisors associated to CL3n(I) are the linear hyperplanes

— > P . There is a canonical isomorphism H (P , 2) — > 2, and any linear

inclusion P — > P

induces an isomorphism

canonical generator for H ( P , 2 ) . [

29

H (P , 2) — > H (P , 2) giving a

(For facts on the topology of P , see Spanier

]).

PROPOSITION B On p " , c.(0

n

(l)) = positive generator of H (P , 2) = cohomology class of a hyperplane.

The proof is by induction on n. Since O n (l)

j = 0 ^ 1 ( I ) , and since taking Chern

classes commutes with restriction, it suffices to prove this for P . But this is obvious. Now consider continuous complex Hie bundles on finite polyhedra.

The complex

projective spaces are classifying spaces for the funtor which assoicates to each finite polyhedron its group of complex line bundles.

This means that, given a complex

line bundle L on a CW conpte X of dimension < 2n, there is a map f : X — > P such that L ——> f* (0_ n (l)).

The map f is unique up to homotopy. Forming the a>

limit as a topological space P

n

= lim P , with the line bundle O a(l) = lim O^i(l) η

P

η

Ir

we can state the following THEOREM C For any polyhedron X there is a 1-1 correspondence between isomorphism classes of continuous complex line bundles on X and homotopy classes of maps from X to p "

86 V. 1.4. A proof of this theorem appears in Spanier [

29 ]. We shall prove a stronger

theorem with this as a corollary later in this chapter. Consider now this theorem in the special case of holomorphic line bundles on analytic spaces.

Suppose the holomorphic line bundle L on the analytic space X to

be generated by a finite number of global sections, ψ ,... ,φ

e H (X, L). By picking

a trivialization of L around any point a, say on a neighborhood U of a, one gets functions φ

„ , . . . , φ ,, which do not vanish simultaneously, and from these one o,U' ' n, U " il

gets a holomorphic map U — > (E

- {0} which in turn defines a holomorphic map

U — > P . The last map does not depend on the trivialization and in this way a Ti

(£•

holomorphic map X

>P

has been defined.

This map induces an isomorphism

L SL-> φ* (0^ 1 (I)) with ψ ,...,φ

backs of the global sections χ , . . . , χ map φ: X —> P

of O n(l) ° n

as the pull-

IP · Conversely any holomorphic

defines a holomorphic line bundle φ ( υ ) . A s i m i l a r analysis in the c a s e w h e r e the columns i., . . . , i

(with i . < . . . < i , ) a r e linearly independent shows that G r a s s (k,n) is covered by open sets W

1

= φ (U.

I

Hc ,

.

), each i s o m o r p h i c , both holomorphically and a s a l g e b r a i c

*Γ"'^

„ ~k(n-k v a r i e t i e s , t o (E One s a y s that kn φ : C

- S

> G r a s s (k, n)

is a p r i n c i p a l fiber bundle, both holomorphically and algebraically, with fiber C

kn

in (E

GL(k,(E).

-S may be thought of e i t h e r as the set of k-tuples of linearly independent v e c t o r s or a s the set of all surjective m a p s from

(E

to C . T h u s G r a s s (k, n) m a y be

thought of e i t h e r a s the set of k-dimensional l i n e a r s u b s p a c e s of (D η

surjective m a p s from

k

or a s the set of

k

(E

t o (D , modulo i s o m o r p h i s m s of (E . Thinking of the G r a s s k m a n n i a n in t e r m s of s u b s p a c e s , denote (E -S by St(k, n), the Stiefel manifold of k - f r a m e s in (E , which i s t o say of k-tuples of linearly independent v e c t o r s in C . T h e n the m a p φ : St (k, η) — > G r a s s (k, η) a s s i g n s to each k - f r a m e the subspace it s p a n s . Note that G r a s s (1, n) is P ^ ' s

in

F

and that G r a s s (k, n) may be thought of a s the space of

1

F"" .

Thinking of G r a s s ( k , n) as k - s p a c e s in (C , it can be r e p r e s e n t e d in another w a y . GL(n, (E) a c t s t r a n s i t i v e l y on the k - s p a c e s . which leaves fixed the k - s p a c e

z,

We will denote by GL(k, n-k, C) the subgroup

, = . . . = z = 0 . The G r a s s m a n n i a n manifold may be

k+1

η

identified, a s a complex manifold on a l g e b r a i c variety, with

GL(n,(E) /GL(k, n - k , C ) .

T h i s shows that G r a s s (k, n) h a s a t r a n s i t i v e group of a l g e b r a i c a u t o m o r p h i s m s . In t e r m s of our p r e v i o u s d i s c u s s i o n , GL(n,C) a c t s by right multiplication on St(k, n)

96

V.2.3.

and this action descends to Grass (k,n). Fixing the usual Hermitian inner product in (E , the unitary group U(n, (E) also acts transitively on the k-spaces. Denoting the subgroup leaving the space defined by ζ

= . . .=z =0 fixed by U(k, n-k, ¢), the Grassmannian may be identified, as differ

-entiable manifold, with U(n, (E) / U(k, n-k). This shows that the Grassmannian is compact. The Grassmannians are in fact projective, and each has a special projective imbedding, called the Plucker imbedding . To get this imbedding,first map St(k, n) —> P

/ g ) -i

as follows : Use homogeneous

(S)-I ^n coordinates x. . , 1 < i, < i, < n. in IP , and map a point of (E - S to the i . . . .L ' - I k - ' ' determinant of the i

. . . , i columns. The action of GL(k,(E)on St(k, n) will change

the homogeneous coordinates by a constant multiple, so there is an algebraic and holomorphic map η

Grass (k, n) — > P ( k We will show that ρ φ (U

.

is an imbedding. First consider ρ restricted to

)= φ (U). This can be identified with (E ^"

( ί; )-1 (C

)_1

= D + (x

, and ρ maps it into

k(n-k) A by thinking of C as the set of all kxn matrices with first k

columns the identity matrix A = (Ι,Α) and p. . . . (A)= determinant of columns L , . . . , i

of A . Since p.

.

, , .(A) = i+j

coordinate of A, this shows that ρ is an imbedding restricted to φ(υ). One can do the same thing on the opens φ (U. everywhere.

. ), from which it follows that ρ is an imbe dding

th

97

V.2.4.

The polynomials defining the Grassmannian under the Pliicker imbedding may be written down explicitly.

again we use homogeneous coordinates

for every k-tuple

For any k-tuple of numbers between 1 and be zero if two of the indices are the same; otherwise let

it equal

, where

is the permutation such

that For any pair of k-tuples

and any

there is a homogeneous polynomial

The Grassmannian is the variety defined by the To prove that the Grassmannian satisfies these relations it suffices to consider the case

and to show that on

the map

has image contained in the zero locus of

where affine coordinates

are used for

98

V.2.5.

Again consider an element and

as a

d e t of c o l u m n s

kx(n-k) matrix,

letting

Then the relation

r e f l e c t s t h e e x p a n s i o n of t h i s d e t e r m i n a n t b y m i n o r s a l o n g t h e

row.

T o s h o w t h a t t h e G r a s s m a n n i a n i s d e t e r m i n e d b y t h e s e r e l a t i o n s it a g a i n s u f f i c e s to c o n s i d e r the p i e c e with

The relations w i l l suffice to define the G r a s s m a n n i a n .

those relations with

a n d m o r e t h a n o n e of

There are by

of t h e s e ,

First consider just

not b e l o n g i n g t o t h e s e t and they m a y b e indexed

a s m a y be the c o o r d i n a t e s

lexicographically

With r e s p e c t to t h i s

ordering, consider the m a t r i x

a s both in

and

r u n o v e r the k - t u p l e s w i t h a t l e a s t two m e m b e r s not

T h i s w i l l be a l o w e r t r i a n g u l a r s q u a r e m a t r i x , w i t h nothing but p o s i t i v e

o r n e g a t i v e l ' s and 2 ' s a l o n g the d i a g o n a l s (the 2 ' s ,

a s w e l l a s the o f f - d i a g o n a l e n t r i e s ,

c o m e f r o m the c o l u m n s g o t t e n by d i f f e r e n t i a t i n g w i t h r e s p e c t to F r o m t h i s it f o l l o w s t h a t t h e v a r i e t y d e f i n e d by the v a r i e t y of d i m e n s i o n

with

i s contained in a n o n - s i n g u l a r

S i n c e t h i s i s t h e d i m e n s i o n of t h e G r a s s m a n n i a n one k n o w s

a l r e a d y t h a t o n e i r r e d u c i b l e c o m p o n e n t of t h e v a r i e t y T o c o m p l e t e t h e proof one m u s t show t h a t t h e v a r i e t y w i l l be enough to p r o v e d i r e c t l y that

i s e x a c t l y the G r a s s m a n n i a n . i s i r r e d u c i b l e ; it c o i n c i d e s with the

99

V.2.6.

Grassmannian. We have already seen that the imbedding

is essentially a graph, and that to each choice of coordinates

with i1*1 place

there is scactly one choice of the remaining coordinates which will put the point in one to than k,

But the quadratic relations F determine this choice: Given suppose that

will express

, as

runs from

with at least two of these greater

Then the relation

in terms of

with at least one more of

set {1, . . . , k} than is the case with is expressed in terms of

in the

One continues in this way until by the quadratic relations.

.th , l place It may seem that this last argument makes the earlier computation of the rank of a Jacobian matrix superflous. Yet this is not the case, since these two arguments together show that the ideal of polynomials vanishing on generated by the

is actually

, and that the Grassmannian is determined by the quadratic

relations in this strong sense.

100

V.2.1.

Theorem M The Grassmannian manifold Grass (k, n) is a non -singular, irreducible , projective algebraic variety. The Plucker map

is an imbedding, and the imbedded Grassmannian is the variety determined by the quadratic relations As we have already suggested, the Grassmannians are to higher dimensional bundles as the projective spaces are to line bundles : Grass (k, n) has on it a universal bundle of rank k. The bundle itself, which is denoted

, may be defined directly in terms

of transition functions. There is the holomorphic map

where St(k, n) is an open subset of the space of all kxn matrices, and Grass (k, n) is the quotient of the action of GL(k,(E) on St(k,n). For open set

there is the

There is a map

which takes a matrix to the matrix made of columns

set

on

Note that

and that these functions are invariant under the action

101

V.2.8.

of GL(k,(C). These define a k-bundle on Grass(k, n), trivialized along the covering In case

the line bundle

Now we shall define n global section of and

has been defined. over Grass(k,n). For

we define

by-

column of A ) .

This will be invariant under the

action of GL(k, (E) and therefore we get holomorphic maps

They are designed to satisfy the transition rules

and hence define global sections

The sections

will

generate over

Theorem N On the Grassmannian manifold Grass (k, n) there is a holomorphic vector bundle of rank k,

with n global sections

which generate each

fiber. These sections form a basis for H°(Grass (k, n), U^). The last part of this theorem will be proven a little later. A more conceptual description of the bundle in a few different ways. First recall that bundle. There are biholomorphic maps

is in order here. It can be done Grass (k,n) is a GL(k,(C)

102

V.2.9.

such that

is of the form

Thus St (k, n) — > Grass (k, n) is a principle bundle with group vector bundle associated to it is just the dual of

and the

st(k,n) is the bundle of

frames in For another description, think of Grass(k, n) as the k-spaces in

and consider

the product

There will be an algebraic subset V of

x Grass (k, n) consisting of pairs (v,B)

such that veB. The map V — > Grass (k, n) exhibits V as a to be

One can see that V must be

identified with St (k, n). This shews that

-bundle which turns out

because the bundle of frames may be is a subbundle of a trivial bundle, and

103 V.2.10.

that U, is the quotient of a trivial bundle. Denoting the locally free sheaf associated to

U

by

Γ (U ), the map

K

K

,„ ^1'--V >

O exhibits U

T(Uk)

> 0

as a quotient .

Consider again the map §)*: St (k, n)

> Grass (k, n). The (Π - bundle φ (U )

is trivial, as one can see directly from the definition of the transition functions of U . By lifting the exhibition of U

as a quotient, one gets a surjective map of trivial bundles

St (k, η) χ St (k, η) x d

kn considering (E as the space of k χ η matrices. This map is of course the identity. A holomorphic section of φ (U ) over St (k, n) is just a holomorphic map f: St (k, n)

> (E . Those sections of ¢) (U,) which lift sections of U,

over

Grass (k, n) are those f which satisfy gf (A) = f (gA) for all g e GL(k, (E) and Ae St (n, k) -

this should be familiar from the computation

of the global sections of CLn(I) on IP . Again one can use Hartog's extension theorem kn to show that such a lifted global section must extend to a holomorphic f : (E

k > C ,

104 V.2.11.

satisfying the same relations under the action of GL(k, ¢ ) . Then a purely algebraic argument shows that a (E-basis for such maps is given by projecting onto the various columns of the kxn matrix, so that J ) 1 , . . . , φ

are a basis for H°(Grass (k, n), U,).

( This completes the proof of theorem N). Suppose given an analytic space X , with a holomorphic (D - bundle L —> X, generated by Then the A.

έ.,...,

(ί) E H (X, L). Suppose that L is trivial along an open U ^ x .

define a surjective bundle map U χ (En

> U χ Ck

which is the same thing as a map U

> St (k, n)

The induced map U — > Grass (k, n) is independent of the choice of trivialization. Theorem O Given any analytic space X, with holomorphic C -bundle L — > X, generated by ib , . . . , φ

e H (X, L) there is a holomorphic map

ib: X — > Grass (k, n) such that L

> i, (U ), )6.= ib (φ.). Conversely, any holomorphic map φ : X —>Grass(k,n)

induces a (E - bundle φ (U,) generated by the global sections

>ί*(Φ.), ie {1, . . . , η } .

This is exactly as in the case of projective space. It follows from this theorem, as in the case of projective space, that the group of holomorphic automorphisms of

Grass (k,n) may be identified with PGL (n, C). We

already saw that this group of automorphisms operates transitively.

105 V.2.12.

One could approach the Grassmannian from an algebraic point of view and develop a parallel theory for algebraic vector bundles on algebraic varieties.

From the fact

that the universal bundle U. is algebraic, and from Chow's theorem, we get another proof of the GAGA result that holomorphic vector bundle theory and algebraic bundle theory will be the same on a projective variety. variety X

For given a bundle L on the projective

> IP , one can find d such that L(d) = L® O1 a (d) will be generated F^ O χ

by its global sections and therefore induced by a holomorphic (whence algebraic) map to the Grassmannian. To investigate the structure of the Grassmann manifolds the Schubert varieties will be introduced here. Again think of Grass (k, n) as the manifold of k-spaces in C , and St (k, n) as the space of k-frames in C . Fix a filtration O c L c LJ= . . . C L = C of subspaces of C , with L. a space of dimension i. For any k-tuple of integers (a , a , . . . , a ) with O < a. < a < . . · Grass (k, n) is a smooth map this determines the dimension of (a„ . . . , a ). A different proof appears in Chern [8 ] . This book also contains a fuller discussion of the rest of the theorem. -mannians appear in Bott [36].

A topological discussion of the Grass

107

V.2.14.

A Schubert variety (a., . . . , a.) with a. > a. , (a = 0) properly contains the Schubert variety

(a , . . . , a. ., a. -1, a

, . . . , a ). Setting

(a^ . . . , a k ) * = (S1, . . . , a k ) - Σ (a^ . . . , a. _χ, a.-1, - - - , ¾ ) a.>a. , ι l-l it is proven in the book of Chern that (a., . . . , a, )

is a complex manifold which

is topologically a cell of real dimension 2(a + . . .+a,)· This gives a cell decomposition of the Grassmannian, as well as information about the singularities of the Schubert varieties. The Schubert varieties c. (U, ) = (n -k -1, . . . , η -k -1, η -k, . . . , η -k) ι

κ

ν

γ

I

i places

for i = 0, . . . , k, are singled out for special attention. c.(U, ), or more properly the cohomology class

c.(U ) in H ( Grass (k, n),Z), is called the universal i 1

Chern

K

class . This terminology will be explained later. These Schubert varieties have a nice interpretation. (J) by the conditions rank

[

'W

+

°

((a., . . . , a, )) is described

J < k + a.., i e {1, . . . , k j

or equivalently rank (matrix of last η - a. - i columns of A) < k - i, i e {1, . . . , k } . Recall the previous notation that ψ„ . . . , ψ

are global sections of U

over Grass(k, n).

108

V.2.15.

From the last description of the Schubert varieties it follows that {points where the sections are linearly dependent}. In particular,

Grass (k, n) {points where are linearly dependent} More properly, one speaks of the cohomology classes defined by these conditions. In case

is

and

is represented by a hyperplane.

More generally, there is the Plucker i mbedding Grass N

o

w

i

s

a

s

may be seen from comparison of the

Grass (k, n) transition functions. Also

defines a global section of

indeed one induced from a section of

over

, so

is

always the intersection of Grass (k, n) with a hyperplane. It should be noted that is usually not a generic section of varieties representing

always have singularities for

In fact, the Schubert while the

intersection of the Grassmannian with a generic hyperplane is non-singular. The Schubert variety representing symbol

deserves special consideration. It has the

109 V.2.16.

(η - k - 1, . . . , η - k -1)

and can be described as the set of k-spaces in (E which are contained in the subspace defined by ζ = 0. This suggests that the Schubert variety may be identified with Grass (k, η-1). In fact the imbedding St(k,n-1) by

A

> St (k,n) > (A; )

induces an isomorphism of Grass (k, n-1) with this Schubert variety. In particular, c ( U , ) is non-singular. In general c. ( U.) has singularities for i f 0, k and the singularities of c. (U, ) are contained in c

( U, ) - these results are contained

in Kleiman [3 7], The simplest Grassmannian which is not a projective space is Grass(2, 4), which 3

"

may be thought of as the lines in JP . It is a 4 -fold, and the Plucker imbedding ρ : Grass (2, 4) — > TP gives it as the hypersurface defined by X

X

12 X 34 " X 3 2 X 1 4 + X 4 2 X 1 3

12 X 34

+ X

=

°

23 X14 " X 24 X13 " °

using homogeneous coordinates x.„, x.„, x.., x„„, x„., x„. in IP . The Schubert varieties have symbols

110 V.2.17.

0 - dimensional { (0,0) 1 - dimensional {(0,1) 2 - dimensional j (0, 2)

(a, D 3 - dimensional {(1, 2) 4 - dimensional {(2, 2) so the Betti numbers are b = b = 1, b = b , = 1 b , = 2. b , = 0. c. (U n ) is 0 represented by the variety Y (x

2 6 4 odd 1 I ) on the Grassmannian and will have one singularity, o4

2 at the point (1, 0, . . . , 0). c ( U ) will be isomorphic to P and defined by + Y ν V (χ χ , x„,)· See Safarevic [ 35] for a discussion of how one uses Grass (2, 4) 3 to determine which surfaces in P have lines on them, and how many. Given a complex manifold M, with a holomorphic (C -bundle L induced by a map (j) : M

> Grass (k, n), 1*

the cohomology classes

2i

(p (c.(U )) e H (M, Z) are called the Chern classes of L.

It follows from this definition that the Chern classes are represented by analytic subvarieties.

Although it is not clear from this definition, the Chern classes depend

only on the isomorphism class the bundle L, not on the map φ . Two main theorems to be discussed later are: Theorem Q Let the complex manifold M be either an affine algebraic variety or a projective variety.

Then the subring of the cohomology ring H* (M, (Q) generated

Ill

V.2.18.

by analytic cohomology classes (that is, cohomology classes of analytic subvarieties) is the subring generated by the Chern classes of holomorphic vector bundles. Theorem H

'even

R Let the complex manifold M be affine algebraic. Then the ring

(M, ¢) is generated by the Chern classes of holomorphic vector bundles.

Both these theorems are some distance away. They require all we have done so far, together with the Bott periodicity theorem and a deep theorem of Grauert, both to be discussed in the next chapters.

112

Chapter Five

Section Three

Chern classes and curvature

In this section we shall discuss vector bundles and their characteristic cohomology classes from the point of view of topology and differential geometry. We shall connect the topological and differential results and then focus on the special case of holomorphic bundles on complex manifolds . For the rest of our notes all our topological spaces will be connected, locally compact, countably compact CW complexes of finite dimension, unless we make special mention to the contrary. Also, our spaces will almost always have the homotopy types of finite simplicial complexes (see Spanier [29] for our topological terminology). The investigation of the characteristic classes of complex vector bundles will make use of the splitting principle : If

E

>X

is a continuous complex vector

bundle, there is canonically constructed another space

Y

with a map

ir : Y — > X

such that (1)

(2)

-

π* : H ( X ) Z ) — > H ' ( Y , Z )

ff*(E)

is a sum of line bundles .

The construction of such that

it*

is an injection

it: Y — > X

is inductive. We first construct

is injective on integral cohomology and

bundle as a direct summand, we do the same thing with

it * (E) = L θ E'

E' — > Y

.

where

ir*(E) L

it : Y — > X

splits off a line

is a line bundle. Then

V.3.2

113

π : Y —s> X has

fiber

with fiber

C , τ P

is constructed as a projective bundle over : Y —> X

, denoted

E —s> X

is the associated bundle of projsctive spaces,

IP(E) . A point of

through the origin of a fiber of

X . If

P(E)

corresponds to a line

E . P(E) may be constructed by letting (D

act on E - {zero section} and then taking the quotient . In this construction such that back

E

L to

P(E)

is provided with a tautological line bundle

restricted to any fiber is

O (-1) . L pd-1

P(E) then assigning to each point of

fiber which it represents. L

L,

is constructed by pulling

P(E) the line through the

is naturally a sub line bundle of IT (E) .

There is then an exact sequence of bundles on P(E)

0 — > L — > π I (E) — >

ff

* (E)/L — > 0

and we want to show that this sequence splits, so will use the fact that

P(E)

f*(E) — > L © τ* (E)/L . We

is a paracompact space, so

τ*(Ε)

admits a

Hermitian metric, as was discussed in Section one of this chapter. Then ir.*(E)/L is identified with the orthogonal complement bundle to

L , and

π* (E)

is the

direct sum. (This argument shows that, on good spaces, exact sequences of bundles always split.) THEOREM S : The integral cohomology of integral cohomology of

P(E) is a free module over the

X , with basis 1 , C 1 (L), [c (L)] , . . . , [ c (L)]

The proof of this theorem comes from the Leray spectral sequence of the fibration

P ( E ) — > X . A discussion appears in Spanier [2¾ .

V.3.3

114

It follows from this theorem that there is a relation

where the

are uniquely determined ,

is called the definition

The

Chern class of the vector bundle

is extended by setting

-The

so defined

for c(E)

The

is used to denote

and is called the total Chern class of Note that in case

E

is a line bundle this definition of

E .

agrees with the

previous one. THEOREM T : (

1

The Chern classes have the properties )

o

n

a bundle

(2)

bundles. Part (1) of this theorem follows from the functoriality of construction of

To get part (2), we first prove : If line bundles then

F(E) .

is a direct sum of From the definition of the Chern classes,

this is equivalent to showing Now

L

is a sub line bundle of

so contains a trivial sub line bundle. This means

that

has a non-vanishing global section

where

is a global section of

does not vanish; then is zero in

1P(E)

and

Let is the union of the

s.

We can write be the open set where

. Now is zero in

V.3.4

115

4

2d

H (U. UU. , Z ) . Continuing in this way,

II(c (L.) - c (L))

is zero in

H (IP(E), Z ) .

Now part (2) will follow from this and the splitting principle. Before moving on, we should mention that the space pull-back of

E

C

E

E

Y—s> X

c ...czE

consists of all possible filtrations

with one-dimensional successive quotients, of the

JL, X

Q. X

over a point

χ e X . This follows from the inductive construction of

J.. X

fiber of

s> X , on which the

splits into a sum of line bundles, is called the flag space of

E —s> X . A fiber of O c E1

Y

THEOREMU: bundle are the

Y .

On the Grassmannian Grass (k,n) the Chern classes of the universal

c.(U, ) as previously defined.

The proof is deferred to the end of Section Four of this chapter. Now we shall restrict our attention to differentiable manifolds and differentiable vector bundles, seeing how to represent Chern classes by differential forms Let

M be a differentiable manifold,

bundle of rank denote by on

E

E

s> M a differentiable complex vector

d . On M there is the sheaf of sections of

E , and the sheaf of complex 1-forms, denoted by

is a map D : E

> T(M)* ® E (E

satisfying D(fe) = df *> e + f De for a function

f

and a section

e

of

E .

E , which we shall also T(M)* . A connection

v-3-5

A connection on a bundle provides a way to differentiate sections of that bundle: If

v

is a tangent vector at sane point, and

e

neighborhood of that point, then the derivative of

is a section of e

E

in the direction

defined in a v

is

j

an element of the fiber of

E

over our point.

Continuing in this vein, suppose that

is a frame of

E

over an

open set (that is, which gives a basis in every fiber). Then a section over this open can be expressed as derivative of

e with respect to

with smooth functions v

, and the

will be

Thus to differentiate a section we pick a frame, differentiate componentwise, then add on a correction term depending on the frame. We can write

>

the

being 1-forms. Then

to the frame Given a connection we define maps

is called the connection matrix with respect

117

V'3

by

The map

satisfies

and is threrfore a linear map of bundles. This map is called the curvature of the connection. In terms of a frame

The matrix of 2-forms the frame

we have

is called the curvature form matrix with respect to . We can compute it in terms of the connection form:

so

The curvature can also be expressed in terms of the Lie derivative (this is because the exterior derivative of forms can be defined in terms of the Lie derivative).

-6

V.3.7

118

If

η, ν are vector fields over an open set and

e

is a section of

E

over that

set then D (D e) - D (D e) - D r _e η ν ν η [η, ν]

is equal to

2 D e(n,v) . We give a proof by computation in local frames:

D (D e.) - D (D e.) - D r ,e. nx v i ν η ι [η, vj ι

= D ([«.(v)e.) - D v(Σ ω..(η) e.) - Σ o\.([n, v]) e.. nVL r p ^y v i]v y i] N L ' i]

Now the relation between exterior differentiation and the Lie bracket is

d (n, v) = η w(v) - vw(n) - w([n, v]) w for a 1-form

w.

Using this in our last expression yields

D (D e.) - D (D e.) - D r e η ν ι ν η ι [η, ν]n ι

= Σ do)..(n, ν) e. - ΣΣ ω., Λ ο). . (η, ν) e.

which is the curvature. (For a discussion of the relation between Lie differentiation and exterior differentiation, see Hicks [16], and the references given there.) Connections can also be defined in other ways, some of which we shall use here. For instance, let

E

differentiable manifold

>M be a complex vector bundle (differentiable) over a M . The total space

with sheaf of tangent vectors

T(E) .

E

is also a differentiable manifold

We denote by

V(E)

the subsheaf of tangent

V.3.8

119

vectors which are linear along the fibers. To see what this means, note that if is trivial,

E =C x M .

Then T(E) = C χ T(M) . A vector field

is called linear if the induced map

V(E) is defined. The restriction

V(E) to the zero section is a sheaf on M , denoted

T(E) > T(M) (Γ ο-section (C

V(E) . The map

induces a map

V(E)

which is surjective.

E—->T(E)

(C —> QT is linear. This definition is invariant

under the trivialization and in this way the sheaf of

E

>T(M) (E

The kernel may be identified with

Horn (E, E)

and there is

an exact sequence

0

>Hom(E,E)—> V(E)->T(M)

>0

V(E) , as a subsheaf of T(M) (C

T(E) , inherits the structure of Lie algebra, as does (E is also a sheaf of Lie algebras.

In this framework, a connection in

φ: T(M) —s> V(E) which (E splits the exact sequence. The curvature of the connection is an alternating map from

T(M) χ T(M) to (D

E

is a map

V(E).

Horn (E,E) , defined by

(C

i ( R ( t , n ) ) = 0 ( [ t , n ] ) - [φ (t), φ (η)]

which is an element of

Horn (E,E) .

This point of view is useful to show the existence of connections for, as we have seen, an exact sequence of vector bundles on a paracompact differentiable

V.3.9

120

manifold always admits a splitting. To show that this definition of connection is equivalent to the previous one, we begin by showing how a connection map D : E

> E * (T(M)*) (C

gives a splitting. If

e , . . .,e ,

D(e.) = Σ oo.. e. , then the map

is a frame for

E

ψ : T(M) — > V(E)

over an open set, and is given by

: t — > ( 0 ) . . ( t ) , t)

under the local isomorphism

V(E) - = ^ H o m ( E , E ) g T ( E ) (C Conversely, given a splitting matrix of 1-forms

φ and choosing a basis

e , ...,e

one gets a

w.. . Some computations in local coordinates show that these

definitions are the same, and that the definition of curvature is consistent. After choosing a frame of 2-forms

Ω

a={e.,...,e

. If the frame is changed to

} the curvature is given by a matrix β= {f , . . . , f

}

with

(f, . . . , f ) = g

(e , . . . , e ) then the change of the curvature matrix is given by

This shows that

det(Q f t ) = det (Ω ) is a well-defined differential form, as is

tr(Q J = tr(Q ) . By locally expressing the curvature in frames, we get an expression

121

d

1

det ( t l d + 27Z7 Ω) = t + c ( f i ) ^ " + . . . + c d (fi)

where

c.(Ti) is a globally defined complex-valued differential form of degree

THEOREMV:

The differential forms

curvature induced by another connection in cohomology class defined by

c.(0)

c.(0) E , then

is the i

are closed. M Ci c.(n ) - c.(Q)

Chern class of

E.

2i .

is the

is exact. The m

H (M, (E) .

We shall give a complete proof of this theorem only in the case where

M

is

a compact differentiable manifold. A proof in the general case appears in Kobayshi-Nomizu [18] . Proceeding with our proof, consider the space of d xd complex matrices, gl(n,(E) . A polynomial function p(gAg

) = p(A) for

ρ : gl(d, (E) — > (E is called invariant in case

A e gl(d, (E) , ge GL(d, (E) . The invariant polynomials form

a ring. Particular examples of invariant polynomials are the functions

p.(A)

defined by det (tI.+A) = I 1 1 H - P 1 ( A ) I 1 1 " 1 + . . . + p n (A) 1 1 and the

s.(A)

defined by

trace (A ) = s.(A)

The ring of invariant polynomials is generated over the

s.(A), If

rank

d,

(E by either the

p.(A)

or

i e {1, . . . , n } .

Ω is the local curvature matrix of a connection on a vector bundle and ρ

is an invariant polynomial, then

E

ρ(Ω) is a globally defined

of

V.3.11

122

differential form. Letting

I

denote the ring of invariant polynomials, we get

a homomorphism

called the Weil homomorphism by

The image of this homomorphism is contained in the ring of closed 2-forms. For it suffices to prove that one always has

Now d trace

Trace

Recall that if the connection matrix is expressed in local coordinates as then the curvature matrix is expressed in the same local coordinates as

so that Now if

g

Trace g

and

gl(n, (E) then as we see by considering terms linear in

in the identity Trace

Trace

V.3.12

123

Extending this to the case where the

are matrices of 2-forms and the

matrix of 1-forms , and using the expression for

g

is a

we get the desired result.

The Weil homomorphism then reduces to a map into the complex cohomology algebra

It remains to show that this map is independent of the connection in the bundle. Let then

be two connection matrices in local coordinates. For each

a connection is defined by Setting

, one gets

For

, set

Trace Trace

, with curvature

i-1 times by a computation similar to that made before. Also,

Trace

Trace Trace

Trace

Therefore

V.3.13

124

Trace (Ω , . . . , Ω ) - Trace (Ω , . . . , O1) = d Γ i Trace (d^ Ω , . . . , Ω ) dt 1

1

U

U



'

t

t

which shows what we wanted. Before showing that the appropriate differential forms represent the Chern classes, some remarks about connections in complex vector bundles are in order. As we have mentioned before, if

E—> M

is a complex vector bundle on a

differentiable manifold, then E

always admits an hermitian m e t r i c . Specializing

to the case of a holomorphic vector bundle on a complex manifold, provided with an hermitian metric, there is always a unique connection D in

(1)

e, f

The connection matrix of

of

E

on an open set .

D is represented by forms of type (1, O) .

To show that such a connection exists, suppose that functions of

such that

d < e , f > = +

for differentiable sections

(2)

E

{g..}

and the hermitian metric is given by

D may be taken as

E

is given by the transition

{h.}.

The connection matrix

w. = h . oh. , which is of type (1, O) and transforms

properly. To show uniqueness, note that a connection which preserves the hermitian structure must satisfy, in its local connection matrix

dh. = w.h. + h. w 1

nad if

w.

11

is of type (1, 0) it must be

1 1

h . oh. .(The connection matrix will be of

type (1, 0) only with respect to holomorphic frames.)

V.3.14

125

The curvature of such a connection is given by

a matrix of forms of type (1,1) . In particular, if a holomorphic line bundle

L , with transition functions

is given, then a hermitian metric consists in a map

such that

and the curvature of the associated connection is the 2-form

Tracing through the co boundary map

is

represented by

in

. To represent this under the deRham isomorphism, we need 1-forms such that

V.3.1E

126

Then

will represent

. Since

so we can take

, and

This shows how to represent the first Chern class of holomorphic line bundles on a complex manifold in terms of the curvature. To extend this, observe that if differentials manifolds, and D , then

is a differentiable map of is a complex vector bundle with connection

inherits a natural connection

differential forms associated to

. Furthermore, the invariant

are pulled back from those associated to

D

on M . If

are connections on bundles,

, there is a natural connection

. The curvature of this connection is given by the matrix

so that

127

V.3.16

Now by using a flag manifold we get the theorem for the Chern classes of holomorphic bundles on the complex manifolds- we must observe that the flag manifold construction will stay in the holomorphic category if we start with holomorphic bundles. In particular, we get the result for the universal bundle on the Grassmannian. Now any differentiable complex vector bundle on a compact differentiable manifold is induced by a differentiable map to a Grassmannian: Any bundle on a compact manifold is generated by finitely many global sections, which induce a map to the Grassmannian just as in the holomorphic case. Thus we get the result for compact differentiable manifolds.

Only slightly more argument gets the result for

differentiable manifolds with the homotopy types of finite polyhedra.

128

Chapter Five

Section Four

Analytic cocycles

We shall now give our final discussion of the way in which an analytic subvariety of a complex manifold defines a cohomology class. We begin by defining the cohomology class of one oriented differentiable manifold imbedded in another X *—;>M

Suppose that the codimension of M , [X] , N—>X

will be in

X

in M

is

i . The cohomology class of

(see Chapter Two, Section 2, and Milnor |23] for a discussion of normal

Since a neighborhood of X

in

the zero section in

N = N - {zero section}

N is isomorphic to a neighborhood

M (see Milnor [23])

H k (M,M-X, Z ) - ^ - > H k (N, N X , Z )

for all

in

H (M, Z ) . We consider the normal bundle of this imbedding,

bundles). N is a real oriented vector bundle of rank i . Let

of

X

k .

THEOREM W : There is a unique class each fiber to the generator of

re H^N, N X , Z )

H (IR , K. - { 0})

defined by the orientation.

We prove this first for the trivial normal bundle orientation. There is the exact sequence

which localizes on

TR. χ X , with the product

V.4.2

129

The map

is an isomorphism for

and is always injective. For

the cokernel is an oriented copy of

implies the existence and uniqueness of

in this special case.

Now we show the uniqueness of

are both classes in

which restrict to the generator on each fiber, a maximal open subset of open

W

not contained in

X U

, which

on which such that

is not N

is trivial over

let

U

X , there is an

W . Let

be the normal bundles. Then the Mayer-Victoris exact sequence

together with the known result for so

U

is all of

shows that

on

X .

The same argument shows that

T exists.

The same argument also shows that The image of

under

I

for

WUU ,

be

V.4.3

130

is called the fundamental class of

X

in

M

and denoted

[X].

We now know how to define the fundamental class of a smooth analytic subvariety of a complex manifold, since everything always has a natural orientation. Now suppose that

X Λ—> M

complex manifold, and that

X=X

is the inclusion of an analytic subvariety in a .

, the singular locus of

H 7 I(M)M-X)=O

for

X

is smooth. Then

η H r '(M,M-X)- 2 : *> H^(M-X

M-X)

> H ^ ^ M , M-X)

η+ 1 < 2i+2 . In particular,

H 2 l (M, M-X)-=^> H 2 l (M-Xj,M-X)

In case

X

no singularities.

has singularities one excises these in turn, continuing until there are In this way the cohomology class of an analytic subvariety is always

defined. Because of its intuitive appeal, we shall also give a definition of the cohomology class of an analytic subvariety in terms of deRham cohomology. On a complex manifold of dimension

VL

η we denote

(M, R) = group of closed real-valued exacts.

C

differential forms, modulo

V.4.4

131

ι H__

°° (M, R ) = group of closed real-valued C differential forms with compact supports, modulo exacts.

deRham's theorems give isomorphisms between

H

(M,R) and the real singular

L)K

cohomology of

M , and a duality between

H

(M, R )

and

H

UK

(M, R ) , by DK, C

(ω, η ) — > J ω Λ η . M If

X ί—3> M is an analytic subvariety of pure codimension

[X] e H_ D (M,R) will be defined as a functional on H L)K

(M,R) .

L)K, C

AJ (M) the compactly supported differential forms on M of

Denote by degree

i , the

j . We shall see that for any

cp e A

(M) ,

J X-X.

is well-defined, and for

n eA

(M)

X-X.

where the integrations are always taken with the natural orientations on Because

φ has compact support we may assume that

with coordinates

(z

. ..,z )

A-I

We set

M is a polydisk in CE n

and that the projection on the first

induces a branched covering, say of degree d , from

X

X-X

n-k

coordinates

onto the polydisk in (C

n-k

n

ω=-r-Σ.

dz. Adz.

a 2-form in the polydisk. For any n-k linearly

independent complex tangent vectors of type

(1, 0) , t , . . . , t

V.4.5

132

Thus one can assume there is

such that

There fore to show that the integral is well-defined it is enough to show that

is finite .

Now if

Y

is any analytic subset of

measure defined by

, then

Y

has measure

. Then for the purposes of measure theory,

a d-sheeted cover of

0

in the is

. Then

and this last integral is certainly finite. It only remains to see t Stokes'theorem proved in

h

a

t

.

This follows from the extended form of Stolzenberg [33] •

We shall not give a proof that all our definitions of cohomology classes coincide. An element of

is called an analytic cocycle if it is in the group

generated by the fundamental classes of analytic subvarieties. We shall use several facts about analytic cocycles, without proof. For more discussion see King [38] and the references given there.

133

If

N is a complex manifold and

t:M

•> N is a holomorphic map then the

f* map in cohomology takes analytic cocycles into analytic cocycles. The cup product of two analytic cocycles is again an analytic cocycle, so H - (M, Έ)

contains an analytic cocycle subring.

If f : M — s > N is a holomorphic map, and variety of pure codimension i , then in case

f

Y*=—s> N is an analytic sub(Y)

has pure codimension

i

[f _ 1 (Y)] = F[Y] if

f

(Y)

is counted with multiplicities.

We can now complete the proof of the theorem in Chapter Five, Section Two, that the Chern classes of the universal bundles on the Grassmannians are represented by Schubert subvarieties. cp , . . . , φ φ ,φ set by

On Grass (k,n) we had the global holomorphic sections

and we claimed that

c.(U, ) was represented by the analytic set where

, . . . , φ -(k-i) are linearly dependent. W . On W

there is an exact bundle sequence

0 — > Ou , k " 1 - 1 hoi where

We denote the complement of this

Q has fiber dimension

> U, — > Q — > 0 k ^

i-1 . Then

c.(U,)=0

restricted to

W , so

c.(U, ) is in the image of H 2 l ( G r a s s (k, n), W ) — > H 2 1 (Grass (k, n))

Now if we can see that

2i H (Grass (k,n), W)

the result up to constant multiples.

is 1-dimensional, then we shall have

V.4.7

134

The singularities of the Schubert variety representing

c.(U,) ,

(n-k-1, . . . , n-k-l,n-k, . . .,n-k ) i places are contained in the Schubert variety representing

c

(U.) ,

(n-k-1, n-k-1, . . .,n-k-1, n-k, . . . , n-k ) i-1 places

and

c.(U, ) - c. ,(U.) is a complex manifold which is topologically a cell. Then

H 2 l (Grass (k,n),W) - ^ - > H 2 l ( G r a s s (k,n) - e

(U. ),W) i+1

and topologically the pair Grass (k, n) - c

(U, ),W i+1 κ Then the cohomology group is one dimensional.

κ

is like the pair

(E ,C - {0} .

Now we know that on Grass (k, n) c.(U ) = η.[ο. (U, )] , and we know that Consider the flag manifold

F(k, n)

>

Grass (k, n) , where

f*(U.)

rh = 1 ·

is topologically

a sum of line bundles, each of which is itself holomorphic. By looking at everything in terms of these line bundles, and using the fact that the cohomology of Grass (k, n) into the cohomology of

F(k, n) , we can see that

THEOREM X : Let the complex manifold

injects

η. = 1 .

M be either a projective variety or

an affine variety. The Chern classes of holomorphic bundles on M are analytic cocycles. Consider first the projective case. Given a holomorphic bundle there is a holomorphic line bundle

L

such that

E®L

E —> M,

is induced by a holomorphic

map to some Grassmannian by arguments similar to those in Chapter Five, Section

V.4.8

135

One. This shows that die Chern classes of

E^L

are analytic cocycles, since they

pull back from the Grassmannian. We shall see in a later chapter that the Chern classes of E of

are in the ring generated by the Chern classes of E ^ L

and those

L . In the affine case the cohomology vanishing theorems to be discussed in Chapter

Eight will show that it is induced by a holomorphic map to a Grassmannian, and we get the same result.

136

Chapter Six Section ι K-theory and Bott periodicity The results in this chapter will be purely topological. Our topological spaces will always be paracompact, locally compact Hausdorff spaces, having the homotopy type of a finite simplicial complex; we shall call such spaces nice spaces. To each nice space

X we will associate a ring

vector bundles on X

K(X) , which will classify the complex

up to a geometrically describable equivalence relation.

The Bott periodicity theorem gives an isomorphism K(X) ® K(W1)-=^

K(X χ P 1 )

This isomorphism will be the essential ingredient in the construction of a natural isomorphism H"'even(X,Q)

K(X)1? (¾ - ^ s >

which will show that, over

φ , the even-dimensional cohomology classes are

combinations of the Chern classes of vector bundles. We begin by describing the ring THEOREMA: Let Suppose

E— > Y

be nice spaces, and

is a vector bundle. Then

First note that if

E

subset, then any section of neighborhood of

X, Y

K(X) .

over

A

homotopic maps .

f? E -^-> f* E .

is a bundle on a nice space E

f , f :X—s> Y

X , and A c X

can be extended to a section of

is a closed E

A . This is proved first in the case of a trivial bundle by the

in a

VI.1.2

137

Tietze extension theorem, then in the general case oy a partition 01 unity. Now let F : X x I

be the homotopy and denote by

g

>Y

the composite map

f X χ I — > X —ί—> Y

The bundles on X x I

F*E

and this defines a section of

and

g E

are isomorphic restricted to

H o m ( F * E , g * E ) over

Xx

X χ {t} ,

{t}, and thus over a

neighborhood. The section will define an isomorphism on a neighborhood. Thus for every

tel

there is a neighborhood

X χ U . This implies that

On a nice space

has an addition

K(X)

X

U of

t

such that

the set of isomorphism classes of complex vector bundles

(E, E') —> E * E'

and a multiplication

is defined as the free abelian group generated by

[E Φ E'] -([E] + [E']) has the property that any map

γ : Vect (X) — > G

with

on

f* E -^s> f*E .

relations

K(X)

F*E -^=-s> g*E

G a group, which is additive, factors uniquely as

(E, E') —> E ® E'

Vect (X)

modulo the

VI. 1.3

138

Vect (X) r

,> K(X)

\ / " G

with

γ'

a group homomorphism.

Elements of represented as

K(X)

[E] - [ E ' ]

are called virtual bundles. Any element of with

will define the same element of

E,E' K(X)

bundles on X . Two bundles

K(X) E

just in case there is a third bundle

may be and G

F

such

that E θ G —s> F θ G This condition is

sufficient,for

[E] - [ F ] = [ E S G ] - [ F e G ]

To see that this is necessary, consider the monoid

Vect (X) χ Vect (X)/diagonal .

This monoid is in fact a group, and there is an additive map

Vect (X) -2—3» Vect (X) χ Vect (X)/Δ by E

and we will have

γ(E) = y(F)

>(E,0)

only if there is

G as above .

In fact, the induced map

y* : K(X)

is an isomorphism.

> Vect (X) χ Vect (X)/Δ

VI. 1.4

139

We know that on a nice space an exact sequence

of bundles always splits. Thus

K(X)

property with respect to maps from satisfying

could have been defined to have the universal Vect (X)

to groups

for exact sequences

If

F

is another bundle then

is also exact, so

K(X)

is actually a ring. A continuous map of nice spaces

induces Onaaring nice homomorphism space X with bundle

E

there is always a map

VI.1.5

140

where

I,

is the tirvial bundle of rank

d

compact, and it is true because sequence, we get of

K(X)

E ΦK

X

d . This would ae immediate if

X

were

has a compact homotopy type. Splitting this

> I , . It follows that

E,F

define the same elements

just in case Ε Θ Γ ^s> F θ 1 d d

for some

d . In other words, the relation defining

K(X)

is stable equivalence .

Because any bundle on a nice space is a quotient of a trivial bundle, we have THE OREM B : On a nice space

X

there is a natural isomorphism

Vect k (X) -^-i> [X, Grass (k, »)] where

[ , ] denotes homotopy classes of maps and

Grass (k,n)

as

η

goe s to infinity.

(Recall that

Grass (k, Grass (k, n+1)

as the Schubert variety representing the top Chern class of the universal quotient bundle

U, . We always pull back

U, .) CO

For example, so

[X,P

»

~ ]—>

Grass (1, )= P

2 H (X, Z )

is the Eilenberg-MacLane space

K(Z, 2)

which is the group of complex line bundles. For details,

see Spanier [ 29] . K-theory began with Bott's computation of the homotopy groups of the complex general linear group. The natural map into the upper left corner

GL(n,(E) induces a string of maps

> GL(n+l,(D)

VI.1.6

141

It is a fact, to be discussed later, that these maps are eventually all isomorphisms. The resulting group

is called the i**1 stable homotopy group of the general linear group, and this is what Bott computed. We now relate this to vector bundles. If then

E

E

is a vector bundle on a sphere

is trivial when restricted to

p

this space is contractible. Therefore any bundle on bundle on along

Sn

and one on

any point, because

arises by taking a trivial and patching them

{two points} with a map THEOREM C : Let

that

X

be a nice space,

X = A U B . Suppose given bundles

A

E^ , Eg

and on

B nice subspaces such A , B and a bundle isomor-

phism

Then there is a bundle

E

on

X

and isomorphisms

The suchisomorphism that This is calledclass the clutching of E depends construction only on for thevector homotopy bundles. class of

g .

142

To construct

E

VI.1.7

we take the quotient of

relation. This turns out to be a bundle. If of

E^ " E g

g^,g^

by the obvious equivalence

are homotopic isomorphisms

, then there is a bundle isomorphism

which gives a clutching of

and

on

I x X , and induces

on

the ends. An argument like that in Theorem A shows that we get the same bundle on either end, namely the restriction of the total clutching. Returning to

, an automorphism of the trivial bundle on

is given, up to homotopy, by an element of This identifies

thus of

with

We consider a slight generalization of this: For a nice space the unreduced suspension of

where the relation

- {two points [

R

X , denote by

X ,

identifies

to one point and

point.

THEOREM D : There is a natural isomorphism

Bott's original form of this theorem was

to another

VI.1.8

143

One knows that compared to

π (GlX(D)) = 0 , «r (GL((C) — > Z . Then for

k

big enough

η we have

Vect

(S 2 ") — >

TL ,

(S211"1) — > 0

Vect

The explicit forms of those isomorphisms are of great interest and we shall discuss this later.

These lead to the isomorphism

K ( S 2 n ) — > Z[X]/ (X-I) 2

, K(S2n+1)—>Z

Again, the explicit form of the isomorphism is of great interest. This also computes the homotopy groups of the infinite Grassmannians. For k

big enough compared to i , it. (Grass (k,»)) — > π

(GL ((C ))

For an elementary proof of the periodicity theorem in the form 1

K(X x P ) -^->

1

K(X) ® K(F )

see Atiyah [ 2 ] , or Bott [ 4 ] . Bott [ 4 ] also contains Bott's original proof. We shall see in the next section how to derive Bott's oroginal statement from the K-theory

statement.

144

Chapter Six Section 2

K-theory as ageneralized cohomology theory

In this section we shall explore the formal aspects of K-theory by putting it in the setting of a general cohomology theory. Let

P denote the category of finite simplicial complexes. A general cohomology

theory is a sequence of contravariant functors:

F

with

: P — ρ - (Abelian groups)

η e Z , which are homotopy invariant and have the following property: For

a polyhedral pair (X, Y) we define

F n (X, Y) as the kernel of

where

collapsed to a point. It is required that there

X/Y

denotes

X

with

Y

be defined natural transformations

6 : F (Y)—s> F

F n (X/Y)—> F n ( p t ) ,

(X, Y ) , called connecting

homomorphisms, such that the long sequence

>

n F

( Y

) -A_> ρ 1 * 1 (X1 Y )

n¥ >

F

\x)—>

Fn+1(Y)—> Fn+2(X,Y)—>

is exact. Thus a cohomology theory is formally like ordinary simplicial cohomology. We shall now build a cohomology theory out of K-theory. First functors will be defined for to

η < 0 then

K

Bott periodicity will be used to extend the definition

η> 0 . We use

P

to denote the category of finite simplicial complexes with base

2 points, and

P

the category of simplicial p a i r s .

For

X

K (X) = Ker ( K ( X ) - > K(St ))

in

+ P , we define

VI

145

where

i :χ —> X

is the inclusion of the case point. Tnere is always a natural

splitting K(X) — > K(X) ® K(X0) Defining a functor point, we let

P

s> P

by

A I—> A = disjoint union of

K (A)= K (A ). Of course

K (A)

is just

A

with a base

K(A) , which we are

working into a formalism. For

X and

Y

in

P

the smash product X Λ Y

is defined by

ΧΛ Y = X χ Y/ (X χ y 0 ) Ll (X0 X Y ) with the natural base point. For as

X

in

P

a suspension, also in

S Λ X (we consider 1 as the base point of S1»

P , is defined

S ). Then

. . . AS1 η times

is isomorphic to

S , and

S A X

is what we get by successively suspending

η times. We denote the suspension of

X

by

S(X) , the η-fold suspension by

s n (X) · For

X

in P we define Kn(X) = K (S ""(X + ))

for η < 0

X

146

THEOREM E: For

VI. 2.3

the sequence

is exact. An element of the kernel of bundle

E

to

restricted to

The

E

may be represented as

and some k . Since it is in the kernel we see that Y , so we may assume that

can be lowered to

on

E

[ E ] - [I K ] E

for some

is stably equivalent

is trivial restricted to

, and

Y .

is

Now we shall define the connecting homomorphisms. THEOREM F: For

there is an infinite exact sequence to the left

By taking suspensions, it will be enough to show that the sequences

(1) (2)

are both exact. Then the rest follows from the previous theorem. To define

CX

we consider the cone over

has a natural base point. Note that

suspension of

X , if

X

is identified in

,

may be identified with the unreduced with

147

The suspension

VI.2.4

is obtained trom

jjy collapsing a copy of

I

to a

point. Then the exact sequence of the preceding theorem shows that

which shows in particular that Since

virtual bundles on

of rank 0 .

, there is a natural isomorphism

The induced map

is

6 . Then there is a commutative diagram

using the identification of

and the fact that proves (1) .

CY

with

obtained from

is contractible. Since the top of the diagram is exact, this

148

VI.2.5

To prove (2), consider the commutative diagram

Here the top row is exact

means The isomorphisms of the vertical arrows are constructed as before, and all maps except

, which is induced by the isomorphisms and the top row, are the natural

ones. We want to compare

with the natural map

Consider the special case where

,

Y = a small arc as pictured. The

general case is clear from this.

Then flap corresponding to

can be identified with a closed hemisphere of

, with a

VI.2.6

149

The identification ot

with

is obtained by observing

that everything but the flap can contract to a point.

On the other hand, by contracting

we get

to a point, to get

(X)

Imbedding everything in we see that in the first collapsing

SnY

arises by collapsing the top hemisphere,

and in the second case it is imbedded in hemisphere. Hence we can factor

as

obtained by collapsing the bottom

VI.2.7

150

where

η

is the n a t u r a l m a p and

a

i s induced by the d i i t e r e n c e between collapsing

the top and bottom h e m i s p h e r e s .

L E M M A : The m a p T h e m a p t h i s induces space

K

t (X)

> 1-t > K

of

I

(X)

into itself Induces a m a p i s m u l t i p l i c a t i o n by

a : S X

>S X.

-1 , for a g e n e r a l

X . T h i s i s a consequence of the following:

LEMMA: bundle over

F o r any m a p

S Y . Then

f:X

> GL

f —s> [E ] - [I ]

, ( D ) , let

E

induces a group i s o m o r p h i s m

> K (S 1 X)

Um [X , GL (n ,(E] n -> »

w h e r e t h e group s t r u c t u r e on the left i s induced from t h a t of

Since the m a p

a

c o r r e s p o n d s in

denote t h e c o r r e s p o n d i n g

[X, G L ( n , (C)]

to

(GL

fl

, (E ) .

> — , t h i s will

e s t a b l i s h the f i r s t l e m m a . We a l r e a d y have a bijection of s e t s

lim [ X , G L ( n , ( E ) ] — > n - =°

K(S1X)

The fact that t h i s i s a group i s o m o r p h i s m follows from t h e homotopy equivalence of the two m a p s GL(n) χ GL(n) given by

> GL(2n)

VI. 2.8

151

with homotopy given by

with n, COROLLARY G: _If Y is a retract of X , there is an isomorphism for all This is a formal consequence of the existence of the connecting homomorphisms. Applying this corollary to a product

A x B , where

base points, one gets a formula for

. For

A

A, B are spaces with is a retract of

A xB

B is a retract of A xB/A . Applying the corollary twice, and for general space X, Y

This shows that the kernel of the natural map

is

and

VI.2.9

152

identified with

. Since the induced map

is zero, this leads to a pairing

which induces a pairing for ordinary spaces

(since

, and similarly for suspensions).

In particular, taking

is given for all

X, Y

Y

n

to be a point

and the periodicity theorem gives

By taking suspensions, we get

Now

is a free abelian group with one generator for all

is even, original formulation.

if

n

In particular

so if

is odd. This is the periodicity theorem in Bott's

n

VI.2.10

153

Because of the periodicity, the definition of by

K (X)

K (X) = K (X) . Then the sequence of functors

can be extended to

η >0

K , η e Z , will have the

formal properties of a cohomology theory. We now want to compare K-theory with ordinary cohomology theory .

154

Chapter Seven Section 1 The Chern character and obstruction theory The goal of this chapter is to compare K-theory and ordinary cohomology. We shall give two different proofs of the basic theorem, one directly involving obstruction theory and the other a more formal proof involving a spectral sequence. At the end of this chapter we shall apply our results to study algebraic cocycles on a projective variety. We shall study the Chern character , a map from Vect (X) which factors through will not factor through

• even to H ' (X,Q)

K(X). The Chern class map from Vect (X) K(X)

to

H'>even(

'^'

because it is not additive. The Chern character, on

the other hand, is additive and induces a ring homomorphism

K(X)

• even P- H ' (X, Q).

We use the splitting principle to define this map. First, for line bundles, the map

L

> 1-( c (L)

is already additive, but not multiplicative. We set

C1(L)2 C1(L)3 Ch(L) = exp (C 1 (L)) = 1 + C 1 (L) + - ^ + ^ + ...

Then for an arbitrary bundle ir*(E) -^-Z- L 1 © . . . Φ L.

E we pass to the flag manifold

π : Y—> X

where

is a sum of line bundles, and set

Ch(E) = ch (L ) + . . . + ch (L.)

This will be expressible in the symmetric polynomials in

c (L ), . . . , c (L.) and

thus in the Chern classes of

H - (X,Q) . The i-homogeneous

part of

ch(E)

is

E . ch(E)

so defined is in

VII. 1.2

155

while

is the i ^ symmetric polynomial in the

the

This shows that, over

are polynomials in the components of

so defined gives an additive map, and one that satisfies . even Thus it induces a ring homomorphism

'

, and thus a

ring homomorphism

THEOREM A : For a nice space

X , the map

is an isomorphism. We first show that if the total Chern class of a bundle

E

is

1 then mE

is

stably trivial for some E. Wa then show that, up to multiples, bundles with given Chern classes can be constructed. We always assume that

X

is a polyhedron, and fix a

triangulation. We start by checking the case of a 2n-sphere. In this case we shall prove the stronger re stilt

where the map is given explicitly by

VII. 1.3

156

In particular,

( n - l ) ! | c (E) for any bundle

E

on S

This will amount to showing that the isomorphism

K (S

) -^-> Z

is given

explicitly by C

[E] I—> 2 S

This is clear on

n(E)tS2J (n-1)!

where every bundle is stably equivalent to a line bundle and

2 ~ 2 2 c : Vect (S ) —s> H (S ,TL) . The general case will follow by induction, using periodocity. Consider the diagram K(S2nxS2)

= K ( S 2 n AS 2 ) © K ( S 2 ) ® K ( S 2 n )

IT(S211XS2) = H'(S211A S2) θ H ' ( S 2 ) 9 H - ( S 2 n )

The third and fourth vertical arrows are isomorphisms by induction, and the first is an isomorphism since

K ( S 2 n χ S 2 ) = K(S 2 n )® K (S 2 ) , H ' ( S 2 n χ S 2 ) = H (S 2 n ) ® H - (S 2 )

and the Chern character commutes with these identifications. Since the indicated splittings are functorial, this gives a proof. COROLLARY B : The only spheres which could admit complex structures are S ,S , and

S .

The proof will use the fact that on a compact complex manifold tangent bundle

T ,

M , with

c (T) [M] = χ (M) . For a proof of this see Steenrod [30],

Combining this with the result just obtained we see that if structure then ( n - l ) ! | 2 , so

S

has a complex

η = 1 , 2 , or 3 .

Of thesepossibilities it is known that

2 S

4 has a complex structure. S

has no

VII.1.4

157

complex structure as can be seen from the Riemann-Roch theorem tor complex surfaces.

It is not known whether or not

S

has a complex structure.

Before going into the proof of the theorem we need a few more remarks about the isomorphism

K(S

with global section

) — > X . Suppose that

φ such that

index of an isolated zero p

with

B

ρ

(the ball in B.

E

is a

bundle on S

d

,

φ is zero at a finite set of points on E . The

is computed by identifying a small neighborhood of ) then using a trivialization of

E

on B

to make

φ give a map 2 n φ : B - {p}-

->s2*-1

The degree of this map (with everything given an orientation induced from that of S

) is the index of

φ at

ρ . Then

c (E)[S. ] = Σ η

Λΐ

index of

ρ

a proof appears in Steenrod [30] . A (E -bundle on S

is given by a map f : S 2 n _ 1 — > GL(n, (E)

thus an element of

π„

. (GL(n, (E)), say this is given by

f>> \ : — > (f\x), . . - , A x ) )

W _. . , This induces a map

2n-l S

0

„2n-l > S

_>

by fl

(x>

If 1 W

φ at

ρ . Again,

VII.1.5

158



T h e d e g r e e of t h i s m a p will be

c

(E.)[S

1

\ .

To see t h i s , note that

can be extended t o a m a p from the n o r t h e r n h e m i s p h e r e into

(E

by identifying the

1 n o r t h e r n h e m i s p h e r e with the closed

2n-ball and setting

χ ? 0 , f (0) = 0 . T h i s will induce a section of on the lower h e m i s p h e r e ,

f

on t h e u p p e r .

π

(GL(N 1 (E))

I x

f (x) = | | x | f (τι—π)

E , trivialized as

for

( 1 , 0, . . . , 0)

It will have an isolated z e r o only at the north

pole, of index equal t o the a p p r o p r i a t e d e g r e e . isomorphism

χ —s> f (x)

In t h i s way we have computed the

s> Z , for

N

big enough.

The proof of the t h e o r e m will now p r o c e e d by induction on t h e d i m e n s i o n of t h e polyhedron. for

Suppose then that

i s a bundle on the polyhedron

q > 0 . C o n s i d e r f i r s t t h e c a s e in which

m a y a s s u m e that

E

On the o t h e r hand, to get 2k

δ

E

ir„,

X =S

2k

r e s t r i c t e d to the 2k-1 E

X , and t h a t

c (E)= 0

2 dim E = dim X = 2k . Then we skeleton of

X

r e s t r i c t e d t o any of the 2k d i s k s

is t r i v i a l E | X = I, . 2k which a r e attached

i s a l s o t r i v i a l , and the c o m p a r i s o n of t h e s e two t r i v i a l i z a t i o n s on

2k-l

gives a m a p

(GL(k, ( I ) ) . Setting

Ot Tjp(e

: S 2k

2k-l

> GL(k, ( I ) , thus an e l e m e n t of

) = the e l e m e n t of

every t i m e a cell i s a t t a c h e d , we get an e l e m e n t 2k

η

τ of

( G L ( k , d ) ) so defined C

2k

(X, π .

2k

C

(X, Z ) . Now 2k across e

np(e

)=0

just in c a s e the t r i v i a l i z a t i o n on

X

(GL(k,(E))) = 2k-1 "

extends

We will show that, at the cohomology l e v e l , (k-1)! η

so t h a t our hypothesis will give modify the given t r i v i a l i z a t i o n of extending o v e r

X

Ε

= ck(E)

( k - 1 ) ! τ» E on X

cohomologous to z e r o . 2k-1

Then we can

to get a t r i v i a l i z a t i o n of (k-1)' E

159

It follows from the previous descri ption of element of

(GL(k, (E)) tnat

it ,

η

, as an

2k C (X, Z) , is given by deg0i

, 2k

E

E

(k-l)l

On the other hand, by the same argument as in the case of defines the cohomology class of

c, (E) so that

X= 2k-l

,

c

—?>deg a_

(k-1)! η _ = c (E) , in cohomology.

We can now dispense with our assumption that is arbitrary and dimension

S

2 dim E = dim X . If

dim E

is odd, then we can in the same way define

a cochain T7 E eC k _ 1 (X, ir 2 k (GL(N, (E)) . Since

η

(GL(N1(E)) = 0

for

N

big

enough, the trivialization always extends in this case. If dim E map

φ: X

is arbitrary and >Grass(r,n)

variety representing

dim X = 2k is even, then E

for some

c, .(U )

Φ:X Now

n, r = dim E . Outside of the Schubert

on the Grassmannian,

bundle of rank r-k ; since codim (c

is induced by a

U

splits off a trivial

(U )) = k + 1 , we may assume that

> Grass - c t ,(U ) , so E = I , Θ E' , E' a bundle of rank k . k+1 r ' r-k ' 2k c, (E) = c, (E') , and the cochain tj eC (X, GL(r, (E)) is the same as

if the trivializations are chosen properly. Thus one always has

(k-1)! n

= c (E) . CJ

Since the chern classes of

E

are those of

71 , k

E' , we are reduced to the previous

case. This completes the first part of the proof.

VII. 1.7

160

COROLLARY C : Suppose there is no torsion in , for

implies that

E

is stably trivial.

Now we must prove that

is surjective. We

will show that, given ,

Then

, there is for some

such that

m .

We first need some knowledge of the homotopy groups of Grass (k,n). It will be sufficient to know that

for

r

big enough. For

(In fact, for a proof.)

(Grass

r

big enough compared to

if

k , one always has

. See Steenrod [30]

VII 1.8

161

Now given

we can construct, lor big enough

such that over

Take

E

to be trivial restricted to

patched together so that on

, and trivial on each attached 2k-cell,

, the induced element of

is

. Then, as we have seen before,

The bundle

E

is defined by a map

Grass ( r , 2 r )

for

r

big enough. The obstruction to extending

,

over

is an element of

(Grass £,2r ))

say

• We will show that (Grass (2r, r))

. Since

the element of

defined by

over this cell, consequently if may be identified as

just in case is a cocycle

extends

will extend to

element of

which is clearly the obstruction to extending the map into the Grassmannian. On the other hand, we can interpret

by noting that

is in the kernel of

, thus identified with an element of Now

may be identified with the element of

induced by the map

VII.1.9

162

pulling back

. Then

This shows that,

being given, we can find

such that E

over

. The obstruction to extending will be an element of

and since

, we get this extension for free.

Now with the map extending over

there will be an obstruction to ,

Once again

element of

defined by

. To

identify this note that we can split, by a privious argument ,

where fiber dimension ,

. This splitting extends automatically to

and both

extend to

. Then

is represented by a trivial bundle when restricted to the obstruction to extending tion to extending

lies in

,

, and so we know that is

. The obstruc-

(Grass (k, ooWstability relation

stability relation . But the map

is zero, since

and the map is the Chern character, so

VII.1. IO

163

2k+2 Vect

(S

"-

)/stability is

tion to extending

[E]

is

z e r o . Hence [ E ]

automatically extends so the obstruc-

6 ch([E] - [E ]) . Since

E

extends,

δ ch

[E]=O

rationally, so the obstruction is (rationally) δ ch (E) = a . 2k+2 Our previous construction shows that we can find E' e Vect(X ) E |X2k+1 so

is trivial, and

c h ( E ' ) = « . Then

[E] - [E' ] will extend to

X

obstruction to the next extension will be

H' ( X 2 k + 2 , Q) ,

ch([E] - [E' ]) = ν in

, then for free over

X

such that

. The rational

6 ch , _u ([E] - [E' ])

and we continue in the

same way. This completes the proof of the theorem. This theorem has an interesting corollary, which we will mention without proof. The result is due to Thorn· COROLLARY E : Given a compact, oriented differentiable manifold r\e H„, (X, Z ) , there exists a manifold f :M ,

>X

such that

M

c (E) of

, compact and oriented, and a map

f+ [M] = m · j), for some

The idea of the proof is to first find a

C

is the Poincare dual of some multiple of

E, g , · • · , g 1 d

g , g„, · · ·, g

where a

X , and

d = dimension of

m e Z .

bundle

E

on X

such that

η • Then for generic E, the subset

Z

C

sections

where

^ e linearly dependent will be a sort of manifold with singularities

and will represent the Poincare dual of

c (E). Then we resolve the singularities

of

Z, which will be fairly simply, and set

M

—>X

will be the natural map.

M„, = the resolving manifold.

164

CHAPTER

q2.

VII.2.1.

SEVEN

The Atiyah-Hirzebruch Spectral

Sequence

W e s h a l l n o w d o m u c h of t h e p r e c e e d i n g m a t e r i a l o v e r in a m o r e f o r m a l

again,

setting.

THEOREM F:

For each finite polyhedron

spectral sequence

X

there is

a

with

and

where

F

KJ(X)

i s d e f i n e d a s t h e k e r n e l of t h e

The spectral sequence is functorial in sequence commute with

X,

map

a n d t h e d i f f e r e n t i a l s of t h e

suspension.

W e d e f i n e f o r e a c h p a i r of i n t e g e r s

p and q, with

Then

(i)

For

e a c h p a i r of p a i r s

there is a natural

map

and

with

VII. 2 . 2 .

1 6 5

(ii)

For

there is a

such that the

is

map

sequence

exact.

Note that

for

for

Then we can set The map is defined by

where

a

i s the natural inclusion and

The e x a c t n e s s mentioned in of a t r i p l e , properties

and

(ii)

0

i s t h e u s u a l m a p of

i s t h e n p a r t of t h e e x a c t

w h i c h i s a f o r m a l c o n s e q u e n c e of t h e of

K-theory.

K-theory.

sequence

cohomology

166

B e c a u s e the data g i v e n s a t i s f y

VII. 2.3.

(i)

and

(ii)

there is,

according

to a p u r e l y a l g e b r a i c t h e o r e m ( s e e C a r t a n - E i l e n b e r g f 7 ] ) a s p e c t r a l s e q u e n c e such that

where

In

particular

a s w e see f r o m Bott

The

differential

periodicity.

167

is given

VII.2.4.

by

A computation like those we have done b e f o r e shows

is the usual

that

coboundary.

Note that for a l l

r,

( b o q u e t of p - s p h e r e s , p o i n t )

and this is

0

Hence is zero for r

and

if

q

even.

is

odd.

for q odd, In p a r t i c u l a r ,

and so

T ihs u sa n a t u rwaill l t rbaen sa f ohroml oa tmi o nr p oh fi s fmu n c t o r s

168

d e f i n e d on p o l y h e d r a , o p e r a t i o n of t y p e

(

for all 3

d

,

p. .

VII.7.5.

It i s t h e r e f o r e a

cohomology

will be a higher o r d e r

cohomology

operation

A n i m p o r t a n t p r o p e r t y of t h e s e o p e r a t i o n s i s t h a t t h e y with

commute

suspension:

For any polyhedron

induced

X,

there is a natural

isomorphism

by

Bott periodicity s i m i l a r l y gives functorial i s o m o r p h i s m s

By suspending

X

spectral sequence,

in

and using these i s o m o r p h i s m s through the we see that the

diagram

K-theory

whole

169

VII.2.6.

c o m m u t e s , so d-j is a stable cohomology operation -- that i s , it commutes with s u s p e n s i o n . Our m a i n r e s u l t will be that if X is a complex manifold with the homotopy type of a polyhedron and ρ ξ Η ( Χ , Ζ ) is an analytic cocycle, then d ? , + , T]=O for all k > l .

Thus t h e r e a r e

topological o b s t r a c t i o n s to a cocycle being analytic. To u s e this r e s u l t we will want m o r e information about the differentials. THEOREM G: T h e r e is a unique n o n - z e r o stable cohomology operation of type ( 3 , ¾ , ¾ ) ,

and s o m e multiple of it is z e r o , so it

always has its image in the torsion p a r t of the cohomology group. F u r t h e r m o r e , all higher o r d e r stable cohomology operations defined on its k e r n e l have finite o r d e r . F o r proof of t h i s , as well as a g e n e r a l d i s c u s s i o n of cohomology o p e r a t i o n s , see Steenrod and Epstein [31], and Steenrod [32]. We shall use the following f a c t s , d i s c u s s e d in t h e s e references:

The unique stable cohomology operation of type

(3,¾,¾)

induces a unique stable cohomology operation of type (3 , Ή, / 2 ¾ , ¾ / 2¾). 3 This is denoted Sq . Actually, for each i>0 there a r e stable cohomology o p e r a t i o n s , called the Steenrod s q u a r e s Sq1: Hq(X,Z/ZZ)

> H q + 1 (X, £ / 2 ¾ ) ,

VII.2.7.

1 7 0

L a t e r w e shall need the factorization

C O R O L L A R Y H:

There are

formula

isomorphisms

S i n c e a l l the d i f f e r e n t i a l s in our s p e c t r a l s e q u e n c e a r e k i l l e d tensoring with

Now

Then

, we get

isomorphisms

by

VII.2.8.

But

is

an i m a g e

of

Hence

A

similar

argument

Note

that the

non-canonical.

We

works

in the

isomorphisms shall

compare

odd

ca.se.

obtained them

in this

to the

way

Chern

are character

isomorphism.

First

for

q

even,

and

the

map

gives

the

sequence gives

note

is

that

given by

isomorphism of a t r i p l e the

the

usual

isomorphism

identifying

explicitly.

gives

the

The

map

coboundary,

coboundary.

f r o m the or

in other

exact words

VII.2.9.

172

At the

given

is

explicitly

of t h e

trivial dch

in

we

have

the

isomorphism

follows:

where

on

,

Thus The

as

level

form

(E) = 0,

P

second

which

extends

E

is

a

bundle

over

on

equivalent

where

corresponds element

will

to live

forever

in the

spectral

all

X.

sequence

case

which

is

to

say

that

THEOREM for such

all

k

to

just

in

I: case

extends

to

in, there

of

satisfies is

in

K(X),

trivial

on

that

higher

order

classes.

just

173

Now w e show

that

shall

it i s

compute

in We

non-trivial.

of

VII.2.10.

a

begin

specific

example

with the

double

and

suspension

Then

if

or

6 and

0

otherwise.

Furthermore,

[29],

where

this

is

to

our

space.

the

cohomology

operation

is

not

Now

the

to

zero

we

will

Spanier

attach

a

has

a

is

the

7-cell cell

get

decomposition

computed Note

explicitly). that

with

map

attach

If

will

be

Spanier we

(see

use

g:

has

homotopically [29]). this

double

degree

trivial,

Then

to a t t a c h

suspension

g e

7

,

2,

of t h e

Hopf

then the

map,

induced

since induces

and

get

map

(see a A.,.

map, The

h:

S ^ -> A ^ ,

induced

map

and

I'M

will be trivial,

VII.2.11.

so

We need another topological fact:

If

a

is a generator

of

, then

where

is the Bockstein h o m o m o r p h i s m .

A p r o o f of t h e

appears

G

i n S p a n i e r [29], e x e r c i s e

in Chapter F i v e .

is the Bockstein h o m o m o r p h i s m induced

Now to s h o w that is

not

dc

i s not z e r o w e n e e d o n l y s h o w

is computed as follows:

is zero m o d

2,

Hence

by

zero.

Hence and

factorization

then

If

c

is in

that

175 VII.2.12.

will represent

Now

and the m a p f r o m

to

not z e r o .

is given by

is

Then

Now we will also u s e There

Hence

is a v i r t u a l vector

to d e n o t e a g e n e r a t o r of

bundle

E

on

such

that

since

If

E

extended over

then there would

such that

then we would have

since

be

But

on a l l G r a s s m a n n i a n s ,

t h e r e i s no e v e n c o h o m o l o g y on the G r a s s m a n n i a n . functorial,

this shows what we

because

Since

is

wanted.

The e x a m p l e t h e n s h o w s that not a l l the d i f f e r e n t i a l s i n t h e Atiyah-Hir zebruch spectral sequence are zero. s i n c e w e h a v e a c l a s s in our example,

In p a r t i c u l a r ,

on a s e v e n - d i m e n s i o n a l

space

for

176

VII.2.13.

It w i l l b e c o n v e n i e n t t o r e c a s t t h e c o n d i t i o n s of t h i s t h e o r e m in a slightly different f o r m . polyhedron.

Consider the

As usual,

X

diagram

All cohomology is integral cohomology; since h a s no

torsion,

can be well defined; for any

so it i s w e l l - d e f i n e d .

The d i a g r a m is

commutative.

last

is a finite

177

VII.2.14.

THEOREM J:

lives forever in the

s e q u e n c e j u s t in c a s e it l i f t s to

such

spectral that

for some

Now w e w i l l d e d u c e conditions on an o r i e n t e d r e a l b u n d l e of e v e n d i m e n s i o n t o b e

T H E O R E M K: vector bundle and

S

X

Let Let

the a s s o c i a t e d

d i m e n s i o n of

X

E.

complex.

be a finite polyhedron,

sphere bundle.

Let

d

T h e n t h e i m a g e of t h e T h o m

A,B

on

X,

element

d(A,B,f)

construction.

sequence.

of

m a p s to

is a difference

and a bundle m a p

a n i s o m o r p h i s m r e s t r i c t e d to a s u b c o m p l e x K(X,Y)

bundle,

be the fiber

such that

T h e t e c h n i q u e of c o n s t r u c t i n g

complex

class

l i v e s f o r e v e r in the A t i y a h - H i r z e b r u c h s p e c t r a l

Given bundles

with

be an associated ball

We must construct

vector

construction:

f: Y,

A -> B

one defines

b y a v a r i a n t of t h e

F i r s t g l u e t o g e t h e r t w o c o p i e s of

which

clutching X

along

Y,

an

is

178 VII.2.15.

T h e n t h e i s o m o r p h i s m of A

on

X^

with

B

on

X^,

A

and

B

to get

splits because there is a projection with the k e r n e l

along

Y

a l l o w s u s to

The exact

clutch

sequence

so w e identify

K(X,Y)

of

Set

For

our p u r p o s e s all this m u s t be g e n e r a l i z e d .

g i v e n a s e r i e s of b u n d l e s a n d

such

Suppose

maps

that

is exact along K(X, Y).

Y.

We will construct

m

179

F i r s t note that,

r e s t r i c t e d to

Y,

VII.2.16.

there is a

splitting

where The

B .J a r e d e f i n e d o n l y o n

is an i s o m o r p h i s m ,

Y.

Now

since each side is

We define

A p r o p e r t y of t h i s c o n s t r u c t i o n w h i c h w e w i l l n e e d i s following:

If

Y,

i s a s e q u e n c e on

i s a s e q u e n c e on is acyclic over pairing

are

X,

s u b c o m p l e x e s of

acyclic over

acyclic over and

Y,

X

as

the

and

and

then the complex is

under

the

180 VII.2.17.

T h e p r o o f of t h i s w i l l b e

omitted.

Now to p r o v e the t h e o r e m , and

we have the bundle

has a tartological section

multiplication

a n d the

e.

This defines an

exterior

map

sequence

i s e x a c t e x c e p t on the z e r o s e c t i o n . the zero section has codimension

2d

Now r e s t r i c t t h i s to in

B,

the

B.

Since

inclusion

m a y be m o v e d a w a y f r o m the z e r o section by a small homotopy, on

s o t h e a b o v e c o m p l e x i s h o m o t o p i c to o n e

exact

and the d i f f e r e n c e construction gives an e l e m e n t

K(B,B2 with trivial maps With this assumption, we construct and

as follows:

Put

given)

with projection homomorphisms.

Similarly,

Now take Take

, locally free,

with projection homomorphisms. .

Now the theorem follows.

This Lemma provides us with a ring structure on

and a

natural map

If class in

is an algebraic subvariety, of codimension will be

.

Now suppose

d , then its

A

is a finite

polyhedron,

is a homotopy equivalence. suppose that

Since

X

has codimension

d

we may

VII. 3. 5.

187

Then we have the diagram

- cohomology

Now as an element of

is

0

on

;

for if

is a resolution, then is exact on

M-X , s

o

o

n

A , , . 2d-l

Thus there is

image of < in Hie class corresponding to , so it comes from

We want to show that the

corresponds to [X] must restrict to .

in 0

To show that it is

let be a regular point, with local coordinates in a neighborhood is defined by

in [X]

188

VII.3.6.

A local computation exactly like that done at the end of the last section shows that

ch. (β[0 x ] ) locally gives the gnerator. J

THEOREM

N.

If

M

is a projective algebraic manifold,

analytic subvariety, then the cohomology class d, [X]

of the Atiyah-Hirzebruch spectral sequence. as a class mod THEOREM O.

2, If

algebraic cocycles in

[X]

X

an

is killed by all the

In particular, considering

3 Sq [X] = 0 . M

is a projective algebraic manifold, then the

H' (M, φ)

are the image of the composite map

Kj 1 0 1 (M) ® Q —>

ch K(M) ® Q -^-3» H' (M, Q ) .

This follows from the proof of the last theorem.

The theorem is also true if

VII. 3. 7.

189

M

is simply a complex manifold of finite homotopy type,

subvariety.

X

The proof involves more technical difficulties--see

an analytic Douady [39] .

The original paper of Atiyah-Hirzebruch, to which we also refer, also has a formulation in the case where M does not have finite homotopy type. But we are most interested in the projective case.

THEOREM P. the torsion in

Let

Consider the following

M be a projective algebraic manifold. Then all of

H 2 (M, Z ) is algebraic.

It will suffice to find a holomorphic line bundle a

is a torsion class .

L

such that

c (L) = Oi,

if

The exact sequence of sheaves



exp 2ij/-l

„~

0 — > mZ — > O1. , — - — - — > O j , — > 1 hoi hoi leads to H' (M, Oh*Ql) — > H 2 (M, Z ) — > H 2 (M, O h o l ) . Vect1,. ,(M) hoi

Since there is no torsion in

H 2 (M, O, .)

this proves the theorem.

We shall show that the generalization of this theorem is false : given k > 1 , there is a projective algebraic manifold

M and a torsion class in

H 2 k (M, Z) which is not algebraic . All that is necessary is to find an a 2-torsion class

a e H 2 k ( M , Z ) such that

construction due to Godeaux and S e r r e .

Sq αφ O .

M with

We will use a

VII. 3. 8.

190

THEOREM Q.

Let

G

be a finite group,

There is a smooth projective variety

M

m

(m-1)

maps of

O

U , then

such that

in

,

and

F

F

.

U = U g U' . geG β

is an open

M

is a complex manifold,

M , then

M/G

Let

Then

U'

be a neighborhood of

G U' = U ' ,

and

M/G

a

has a natural

To define that structure locally, let

= x}.

G

χ e M

and

χ ,

looks locally like

x

χ

U'/G . x

But we have seen how

If

M

variety:

Let

imbedding. under

U/G

has an analytic structure,

χ

is a projective variety then L —> M

M/G

will also be a projective

be a holomorphic line bundle which gives a projective

By passing to

G ,

M/G = W . Q

U

a finite group of biholomorphic

finite group of analytic automorphisms of

={geG:gx

If

The reader can prove this himself.

We use this result to show that if

G

may be

has naturally the structure of an analytic space

Γ (U/F, O) = Γ (U, O)

analytic space structure.

M

.

C

U/F

equivalent to

K ( Z , 2) χ K(G, 1).

The construction begins with the following fact: neighborhood of

an integer > 1 .

which is

the product of Eilenberg-Maclane spaces, taken to have dimension

m

® geG

g*L

we may assume that

so that it defines a holomorphic line bundle Now if

F

is a coherent sheaf on

W ,

L L'

is invariant on

we claim that

^d

H (W, F ® L'

) = 0,

take a covering

{U|

for of

q > 0 , d W

such that

big enough . π

cohomologically trivial. See Gunning-Rossi [13] .

(U ) η e Z

To prove this, on

M

is always

( {UJ- , F ® L '

)

VII. 3. 9.

191

q

defines

it* η £ Z ( it

q > 0

F ι» L'

Setting

) , δθ< = n .

N

(^

a' =^~^)

and for

d

0

g*a ) ,

big enough,

such that

a' e c q "

( {U a } ,

This proves the vanishing theorem.

Now given a finite group for

) ,

a-1 -1 ®d H a e C ( { IT U ^ , π* F ® L )

there is

δα = π*η .

{uj- , it* F ® L

G , it has a representation in

G L (N+1, C ),

big enough, such that the induced representation in

is faithful.

Then we can construct a variety

P /G ,

P G L (N+1, (E)

with a projective

imbedding. FN

' ΪN P

/G

JL_>

P

n

Our previous construction showed that we can pick the imbedding of

φ

that

P .

η Let

S

(hyperplane section) = sum of

φ

S = it (Fix G)

is algebraic.

# (G) hypersurfaces in

be the image of the fixed point set of

For a given

things so that codimension

S

G, in

and

m,

P /G

is

Let generic

to get one in L c P

> m .

We could do

be a linear subvariety of dimension

N+m.

L,

N L ΠP / G P /G - S

G

GL (m (N+-1) ) .

L ΠS =φ

(note

G.

we can always arrange

this by taking a direct sum of several faithful representations of G L (N+1)

is smooth. )

so

is nonsingular, of dimension

m

For

in

VII. 3.10.

192

so -1 π

L Π P /G

is a smooth variety of dimension

N (L Π P /G) = X

is a smooth variety of dimension

invariant under the operation of of

N-m

m .

hypersurfaces in

G P

on

P

N

,

Also m .

X

will be

and it will be the intersection

N

It follows from the Lefschetz theorem that the map

X I-> in an

(m-1)

PN

equivalence, that is, that

π. (X)

> Ή. (P N )

1

is an isomorphism for

1

i
p " = K (Z , 2 ) ,

and the fact that

we see that there is an

m -1

v. (P n ) = 0 ,

for

0 < i < 2n ,

equivalence

X -—> K ( Z , 2 ) . Now we will show that the variety to

K(Z:, 2) x K(G,1) .

a map, an be induced by

m-1

First note that

equivalence,

L' , the map from

we have a diagram

M = X/G

M

is

(m-1)

X -—> K ( Z , 2)

equivalent

is induced by

> K (Z, 2); let the map from X

be induced by

TT*L' = L .

M Then

1 9 3

VII. 3. Π.

(m-1) equivalence X

>

K (Z, 2) id

>

M

Over the space

K (G, 1) = B G

P , G

on which

space

Y ,

G

K (Z, 2)

there is a universal principal

operates freely with

B G

G

as a quotient.

bundle, For any

there is an equivalence between

G - bundles over

Y «—>

[Y, B ]

The equivalence is obtained by pulling back

P ,

which is a contractible

G

space (one could take

P_ = simplex with

#(G)

vertices).

Then we get

G

a diagram

X

> K (Z, 2) x P

M

> K (Z, 2) x B^ G

where both the top and bottom maps are

m-1

equivalences.

This proves

the theorem. Now one knows from the Whitehead theorem (see Spanier [29] ) H 1 (M, Z ) -^-s> H1 ( K (Z, 2) x

B_,,

Z)

194

for

VII. 3.12.

In particular,

is a direct summand of

for To make our computation we shall use facts about the squaring operations, found in Spanier [29] or Steenrod [32] . Now take

We have

so is

For , generated by an element of degree one.

Now for large

contains

as a

direct summand.

has

3

generators,

and

Now Also

since

for

deg

u .

195

since

Since

deg

VII. 3.13.

u .

Thus

corresponds to a 2-torsion class in

that there is a 2-torsion class in

,

we see

which is not complex analytic.

196

Chapter Eight

Section 1

Stein manifold theory

This chapter continues the study of vector bundles, now from an analytic point of view.

Our results will tend to compare what can be done continuously

on a complex manifold with what can be done holomorphically.

For this purpose

some study of Stein manifolds will be necessary. A Stein manifold is a complex manifold with a strictly plurisubharmonic exhaustion function; that is, a complex manifold

M

+ there is a function

τ : M -> IR , which is

is compact for all

α > 0 , and

means that in any patch of the matrix

δ2τ „ —

Oz, δ ζ

M

will be Stein just in case 2

τ

-

C , such that

1 τ

([0, a ] )

is strictly plurisubharmonic.

with local coordinates

will be positive definite.

This

ζ , . . . , ζ.

This is invariant under

J

holomorphic change of coordinates. A Stein manifold is a special type of Stein analytic space. defined in a similar, but more complicated manner. here.

These can be

This will not be discussed

See Lelong p i ] . The two basic theorems of Stein manifold theory, which have already been

mentioned in Chapter Two, are THEOREM B. M , then

Jf_

F

H q (M, F ) = 0

is a coherent analytic sheaf on a Stein manifold for

q > 0.

For the statement of theorem A we must know that an essentially unique topology can be defined on sheaf

H (M, F)

for any complex manifold

E . See Gunning-Rossi [13]. If

τ: M — > IR+ , let

M

and coherent analytic

is Stein with exhaustion function

VIII

197

M [r] = τ

( (O, r) ) .

It will also be a Stein manifold.

THEOREM A. The map

H° (M, F) -> H° ( M [r], F)

has dense image.

This is a generalization of the familiar Runge theorem in one variable.

A

consequence of this is the THEOREM A'.

On the Stein manifold

M,

F

is generated by its

global sections.

A simple consequence of Theorem

B is the

THEOREM C. On a Stein manifold the natural map 1 Vectf

(M) -> Vect

1 (M) top

is an isomorphism. 1 Vect

There is a natural isomorphism sequence

0 -s> Z

> 0

—**• 0*

2 (M) ^> H ( M, K ).

—-> 1

The exact

leads to the exact sequence

H (M, 0) -> H1 (M, 0*) -> H 2 (M, X) -> H 2 (M, 0) 11

1

V e c t

hoi

and the extreme terms vanish by Theorem B. COROLLARY D. Every element of

This proves the theorem.

H (M, ^ )

on a Stein manifold

M

is represented by a holomorphic divisor. The divisor defined by a global section of a line bundle will define its Chern class. We shall discuss the theorem of Grauert which generalizes this theorem to bundles of arbitrary dimensions.

We shall also generalize the following

VIII. 1.3.

198

THEOREM F.

Let

M

be a Stein manifold.

the class of holomorphic maps of homotopy.

M

to_

Denote by

, modulo holomorphic

Then the natural map

is an

isomorphism. A direct holomorphic homotopy between two

holomorphic maps

consists of an holomorphic map with

Two maps

f,g

homotopic if there is a sequence

are holomorphically with

and

directly holomorphically homotopic. To see the significance of this theorem, note that so

is a

On the other hand, there is a natural map , inducing an element of

by the deRham isomorphism.

We'll see that this is an isomorphism, so that

all of the first cohomology can be realized holomorphically. To prove the theorem we must use this result, which can be proven directly.

Denote by

the sheaf of holomorphic

p

forms on

M.

Then there is the complex of sheaves.

THEOREM G. (holomorphic de Rham theorem) From this and Theorem B it follows that

This sequence is exact.

199

{ closed holomorphic

VIII. 1. A

q-forms}/{exact holomorphic

q-forms}

We'll do this after doing Theorem F. An element of thus of

defines an element Since

Now set, picking

w

w

is integral,

of for any I-cycIe

M,

where the integral is over any path from

to

z .

Then

f

holomorphic, and defines the appropriate class.

This shows that

is

is surjective. To show injectivity, suppose exists on This shows

in

.

M , and defines

Then

log f = g

by

in

To prove the holomorphic de Rham theorem, let of differentiable, complex valued

p-forms on

Two the decomposition the commutative diagram of sheaf complexes

M .

denote the sheaf Recall from Chapter We have

200

VIII. 1. 5.

and the top row is known to be exact.

Given

a neighborhood of

,

of

with

E.

for the fiber of E over

O, and set

210

Then

VIII. 2 . 8.

is a holomorphic f r a m e for

E.

W e w i l l m a k e a n o t h e r a p p l i c a t i o n of t h e s a m e If

M

is a complex manifold,

technique.

two holomorphic bundles

and

o n a r e s a i d to b e d i r e c t l y h o l o m o r p h i c a l l y h o m o t o p i c if t h e r e i s holomorphic bundle

E

on

unit disk in

CD) s u c h

T w o b u n d l e s a r e h o l o m o r p h i c a l l y h o m o t o p i c if t h e y c a n b e b y a c h a i n of h o l o m o r p h i c a l l y h o m o t o p i c

THEOREM W: homotopic bundles are

d i s k of r a d i u s

that

connected

bundles.

On a Stein manifold

holomorphically

isomorphic.

B y i n t e g r a t i n g the h o l o m o r p h i c v e c t o r f i e l d (=

a

1/2) w e g e t a r e s t r i c t e d

on flow

w h i c h w e c a n l i f t to a l i n e a r - a l o n g - t h e - f i b e r s

flow

which induces an isomorphism

W e r e m a r k t h a t t h e t r i v i a l i t y of a n y h o l o m o r p h i c v e c t o r o n t h e b o u n d e d p o l y d i s k i s a s i m p l e c o n s e q u e n c e of t h e o r e m

W.

bundle

VIII

211

COROLLARY X: closed polydisk in (L

Let U be an open neighborhood of the given by

holomorphic vector bundle.

Let E -» U be a

Then E is holomorphically t r i v i a l

on the open polydisk given by F o r some s m a l l

[ z. [ < 1.

| z . [ < 1.

e > 0 t h e r e is defined a holomorphic m a p F : D1+f

x {|z.|

U

by F (w, z) = (wz). Then F n takes the open polydisk to a point and F . is the injection of the disk.

Since

F ^ E and F''"E a r e holomorphically

homotopic b u n d l e s , this p r o v e s the c o r o l l a r y . Now an approximation a r g u m e n t will allow us to deal with open polydisks. THEOREM Y: E v e r y holomorphic v e c t o r bundle on the polydisk

{] ζ. | < 1} in_