This volume offers a systematic treatment of certain basic parts of algebraic geometry, presented from the analytic and
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Contents
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Bibliography
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: 742968 I.S.B.N.: 0691081514
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 19711972.
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.
Ktheory and Bott periodicity
136
Section 2.
Ktheory as a generalized cohomology theory
144
Section 1.
The Chern character and obstruction theory
154
Section 2.
The AtiyahHirzebruch spectral sequence
164
Section 3.
Ktheory 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 complexvalued 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
complexvalued 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 semicontinuous 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 nonsingularity 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 nonsingular 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) — > XYis 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 Omodules 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 nonsingular 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 nonsingular 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 {Wdivisors}
which is an injection, and in fact exhibits the group of Wdivisors 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 mth 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 Cvector 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 0modules, 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 nonzero 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+Ιtuple
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 nonabelian 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 Balgebraic 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. GunningRossi [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,
complexvalued
oneforms, 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 sforms
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 1variable 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 nonzero
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 dsheeted 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 GunningRossi [
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 dsheeted 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 pseudopolynomial
The projection to the z' hyperplane exhibits V as a dsheated 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 n1 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 ClI 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 singlevalued 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 Zariskidense 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 XD
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 nonempty 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 nonzero 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 nonzero One proves that
module then if faithfully flat over
is nonzero. 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 AltmanKleiman [ 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 GunningRossi [ 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(dl))—>H°(pn,F(d ))—>H°(Pn"1,F(d)pnl) —>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^,ηΙΊ 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 11 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(nk 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 ktuples 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 kdimensional 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 ktuples 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, nk, 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 kspaces. Denoting the subgroup leaving the space defined by ζ
= . . .=z =0 fixed by U(k, nk, ¢), the Grassmannian may be identified, as differ
entiable manifold, with U(n, (E) / U(k, nk). 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(nk) 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 ktuple
For any ktuple 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 ktuples
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(nk) 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 kbundle 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 kspaces 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 (Ebasis 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 kspaces in C , and St (k, n) as the space of kframes 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 ktuple 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 ll 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 nonsingular. 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 kspaces in (E which are contained in the subspace defined by ζ = 0. This suggests that the Schubert variety may be identified with Grass (k, η1). In fact the imbedding St(k,n1) by
A
> St (k,n) > (A; )
induces an isomorphism of Grass (k, n1) with this Schubert variety. In particular, c ( U , ) is nonsingular. 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 pd1
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 nonvanishing 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 pullback of
E
C
E
E
Y—s> X
c ...czE
consists of all possible filtrations
with onedimensional 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 1forms, 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
v35
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 1forms. 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 2forms 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 1form
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 οsection (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 1forms
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 2forms
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 welldefined 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 complexvalued 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 KobayshiNomizu [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 2forms. 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 2forms and the
matrix of 1forms , 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
i1 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
"η
'
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 2form
Tracing through the co boundary map
is
represented by
in
. To represent this under the deRham isomorphism, we need 1forms 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,MX, 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 MayerVictoris 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)MX)=O
for
X
is smooth. Then
η H r '(M,MX) 2 : *> H^(MX
MX)
> H ^ ^ M , MX)
η+ 1 < 2i+2 . In particular,
H 2 l (M, MX)=^> H 2 l (MXj,MX)
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 realvalued exacts.
C
differential forms, modulo
V.4.4
131
ι H__
°° (M, R ) = group of closed realvalued 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 XX.
is welldefined, and for
n eA
(M)
XX.
where the integrations are always taken with the natural orientations on Because
φ has compact support we may assume that
with coordinates
(z
. ..,z )
AI
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
XX
nk
coordinates
onto the polydisk in (C
nk
n
ω=rΣ.
dz. Adz.
a 2form in the polydisk. For any nk 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 welldefined 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 dsheeted 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 (ki) 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 ^
i1 . 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 1dimensional, then we shall have
V.4.7
134
The singularities of the Schubert variety representing
c.(U,) ,
(nk1, . . . , nkl,nk, . . .,nk ) i places are contained in the Schubert variety representing
c
(U.) ,
(nk1, nk1, . . .,nk1, nk, . . . , nk ) i1 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 Ktheory 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 evendimensional 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 EilenbergMacLane space
K(Z, 2)
which is the group of complex line bundles. For details,
see Spanier [ 29] . Ktheory 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]/ (XI) 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 Ktheory
statement.
144
Chapter Six Section 2
Ktheory as ageneralized cohomology theory
In this section we shall explore the formal aspects of Ktheory 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 Ktheory. 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 ηfold 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)
> 1t > 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 Ktheory with ordinary cohomology theory .
154
Chapter Seven Section 1 The Chern character and obstruction theory The goal of this chapter is to compare Ktheory 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 ihomogeneous
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 2nsphere. 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 (n1)!
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 RiemannRoch 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
2nl S
0
„2nl > S
_>
by fl
(x>
If 1 W
φ at
ρ . Again,
VII.1.5
158
9η
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
2nball 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 2k1 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
2kl
gives a m a p
(GL(k, ( I ) ) . Setting
Ot Tjp(e
: S 2k
2kl
> 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))) = 2k1 "
extends
We will show that, at the cohomology l e v e l , (k1)! 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 . 2k1
Then we can
to get a t r i v i a l i z a t i o n of (k1)' 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
(kl)l
On the other hand, by the same argument as in the case of defines the cohomology class of
c, (E) so that
X= 2kl
,
c
—?>deg a_
(k1)! 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 rk ; 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 ' rk ' 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
(k1)! 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 2kcell,
, 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 AtiyahHirzebruch 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
Ktheory.
Ktheory.
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
Ktheory
whole
169
VII.2.6.
c o m m u t e s , so dj 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
noncanonical.
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
nontrivial.
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
7cell 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 AtiyahHir 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
MX , s
o
o
n
A , , . 2dl
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 AtiyahHirzebruch 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 difficultiessee
an analytic Douady [39] .
The original paper of AtiyahHirzebruch, 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 2torsion 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
(m1)
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 EilenbergMaclane 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 GunningRossi [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
) ,
a1 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
Nm
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
(m1)
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
m1
First note that
equivalence,
L' , the map from
we have a diagram
M = X/G
M
is
(m1)
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.
(m1) 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
m1
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 2torsion class in
that there is a 2torsion 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 GunningRossi [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
qforms}/{exact holomorphic
qforms}
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 IcycIe
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
pforms 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_