Complex Algebraic Geometry

239 Pages·2011·1.446 MB·English

by  Gallier J., Shatz S.

Complex Algebraic Geometry
∗ ∗∗
Jean Gallier and Stephen S. Shatz
∗
Department of Computer and Information Science
University of Pennsylvania
Philadelphia, PA 19104, USA
e-mail: [email protected]
∗∗
Department of Mathematics
University of Pennsylvania
Philadelphia, PA 19104, USA
e-mail: [email protected]
February 25, 2011
2
Contents
1 Complex Algebraic Varieties; Elementary Theory 7
1.1 What is Geometry & What is Complex Algebraic Geometry? . . . . . . . . . . . . . . . . . . 7
1.2 Local Structure of Complex Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Local Structure of Complex Varieties, II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4 Elementary Global Theory of Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2 Cohomology of (Mostly) Constant Sheaves and Hodge Theory 73
2.1 Real and Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.2 Cohomology, de Rham, Dolbeault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3 Hodge I, Analytic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.4 Hodge II, Globalization & Proof of Hodge’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 107
2.5 Hodge III, The K¨ahler Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
2.6 Hodge IV: Lefschetz Decomposition & the Hard Lefschetz Theorem. . . . . . . . . . . . . . . 147
2.7 Extensions of Results to Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3 The Hirzebruch-Riemann-Roch Theorem 165
3.1 Line Bundles, Vector Bundles, Divisors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
3.2 Chern Classes and Segre Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
3.3 The L-Genus and the Todd Genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
3.4 Cobordism and the Signature Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
3.5 The Hirzebruch–Riemann–Roch Theorem (HRR) . . . . . . . . . . . . . . . . . . . . . . . . . 232
3
4 CONTENTS
Preface
This manuscript is based on lectures given by Steve Shatz for the course Math 622/623–Complex Algebraic
Geometry, during Fall 2003 and Spring 2004. The process for producing this manuscript was the following:
I (Jean Gallier) took notes and transcribed them in LATEX at the end of every week. A week later or so,
Steve reviewed these notes and made changes and corrections. After the course was over, Steve wrote up
additional material that I transcribed into LATEX.
Thefollowingmanuscriptisthusunfinishedandshouldbeconsideredasworkinprogress. Nevertherless,
given that Principles of Algebraic Geometry by Griffith and Harris is a formidable source, we feel that the
material presented in this manuscript will be of some value.
Weapologizeforthetyposandmistakesthatsurelyoccurinthemanuscript(aswellasunfinishedsections
andevenunfinishedproofs!). Still, ourhope isthatby its “freshness,”this workwillbe ofvalueto algebraic
geometry lovers.
Please, report typos, mistakes, etc. (to Jean). We intend to improve and perhaps even complete this
manuscript.
Philadelphia, February 2011 Jean Gallier
Acknowledgement. MyfriendJeanGallierhadtheideaofattendingmylecturesinthegraduatecourse
in Complex Algebraic Geometry during the academic year 2003-04. Based on his notes of the lectures, he
is producing these LATEX notes. I have reviewed a first version of each LATEX script and corrected only the
most obvious errors which were either in my original lectures or might have crept in otherwise. Matters of
style and presentation have been left to Jean Gallier. I owe him my thanks for all the work these LATEXed
notes represent.
Philadelphia, September 2003 SSS
5
6 CONTENTS
Chapter 1
Complex Algebraic Varieties;
Elementary Local And Global Theory
1.1 What is Geometry & What is Complex Algebraic Geometry?
The presumption is that we study systems of polynomial equations
f (X ,...,X ) = 0
1 1 q
. . .
. . . ( )
 . . .
†
 f (X ,...,X ) = 0
p 1 q
where the f are polynomials in C[X ,...,X ].
j 1 q
Fact: Solvingasystemofequationsofarbitrarydegreesreducestosolvingasystemofquadraticequations
(no restriction on the number of variables) (DX).
What is geometry?
Experience shows that we need
(1) A topological space, X.
(2) There exist (at least locally defined) functions on X.
(3) More experience shows that the “correct bookkeeping scheme” for encompassing (2) is a “sheaf” of
functions on X; notation .
X
O
Aside on Presheaves and Sheaves.
(1) A presheaf, , on X is determined by the following data:
P
(i) For every open U X, a set (or group, or ring, or space), (U), is given.
⊆ P
(ii) If V U (where U,V are open in X) then there is a map ρV : (U) (V) (restriction) such that
⊆ U P → P
ρU =id and
U U
ρW =ρW ρV, for all open subsets U,V,W with W V U.
U V ◦ U ⊆ ⊆
(2) A sheaf, , on X is just a presheaf satisfying the following (patching) conditions:
F
7
8 CHAPTER 1. COMPLEX ALGEBRAIC VARIETIES; ELEMENTARY THEORY
(i) For every open U X and for every open cover U of U (which means that U = U , notation
⊆ { α}α α α
U U ), if f,g (U) so that f ↾U =g ↾U , for all α, then f =g.
α α α
{ → } ∈F S
(ii) For all α, if we are given f (U ) and if for all α,β we have
α α
∈F
ρUα∩Uβ(f )=ρUα∩Uβ(f ),
Uα α Uβ β
(the f agree on overlaps),then there exists f (U) so that ρUα(f)=f , all α.
α ∈F U α
Our is a sheaf of rings, i.e, (U) is a commutative ring, for all U. We have (X, ), a topological
X X X
O O O
space and a sheaf of rings.
Moreover,ourfunctionsarealways(atleast)continuous. Picksomex X andlookatallopens,U X,
∈ ⊆
where x U. If a small U x is given and f,g (U), we say that f and g are equivalent, denoted
X
∈ ∋ ∈ O
f g, iff there is some open V U with x V so that f ↾ V =g ↾V. This is an equivalence relation and
∼ ⊆ ∈
[f]= the equivalence class of f is the germ of f at x.
Check (DX) that
lim X(U)=collection of germs at x.
O
−U∋→x
The left hand side is called the stalk of at x, denoted . By continuity, is a local ring with
X X,x X,x
O O O
maximal ideal m = germs vanishing at x. In this case, is called a sheaf of local rings.
x X
O
In summary, a geometric object yields a pair (X, ), where is a sheaf of local rings. Such a pair,
X X
O O
(X, ), is called a local ringed space (LRS).
X
O
LRS’s wouldbe useless without a notionof morphismfrom one LRS to another, Φ: (X, ) (Y, ).
X Y
O → O
(A) We need a continuous map ϕ: X Y and whatever a morphism does on , , taking a clue
X Y
→ O O
from the case where and are sets of functions, we need something “ .”
X Y Y X
O O O −→O
Given a map ϕ: X Y with on X, we can make ϕ (= direct image of ), a sheaf on Y, as
X ∗ X X
→ O O O
follows: For any open U Y, consider the open ϕ−1(U) X, and set
⊆ ⊆
(ϕ )(U)= (ϕ−1(U)).
∗ X X
O O
This is a sheaf on Y (DX).
Alternatively, we have on Y (and the map ϕ: X Y) and we can try making a sheaf on X: Pick
Y
O →
x X and make the stalk of “something” at x. Given x, we make ϕ(x) Y, we make and define
Y,ϕ(x)
∈ ∈ O
ϕ∗ so that
Y
O
(ϕ∗( )) = .
Y x Y,ϕ(x)
O O
More precisely, we define the presheaf ϕ on X by
P Y
O
ϕ (U)= lim (V),
P Y Y
O O
V⊇−ϕ→(U)
where V ranges over open subsets of Y containing ϕ(U). Unfortunately, this is not always a sheaf and we
need to “sheafify” it to get ϕ∗ , the inverse image of . For details, consult the Appendix on sheaves
Y Y
O O
and ringed spaces. We now have everything we need to define morphisms of LRS’s.
(B) A map of sheaves, ϕ: ϕ , on Y, is also given.
Y ∗ X
O → O
It turns out that this is equivalent to giving a map of sheaves, ϕ: ϕ∗ , on X (This is because
Y X
e O → O
ϕ and ϕ∗ are adjoint functors, again, see the Appendix on sheaves.)
∗
ee
1.1. WHAT IS GEOMETRY & WHAT IS COMPLEX ALGEBRAIC GEOMETRY? 9
In conclusion, a morphism (X, ) (Y, ) is a pair (ϕ,ϕ) (or a pair (ϕ,ϕ)), as above.
X Y
O −→ O
When we look at the “trivial case”’ (of functions) we see that we want ϕ to satisfy
e ee
ϕ(m ) m , for all x X.
ϕ(x) ⊆ x ∈ e
This condition says that ϕ is a local morphism. We get a category .
e LRS
After all these generalities, we show how most geometric objects of interest arise are special kinds of
LRS’s. The key idea is teo introduce “standard” models and to define a corresponding geometric objects,
X, to be an LRS that is “locally isomorphic” to a standard model. First, observe that given any open
subset U X, we can form the restriction of the sheaf to U, denoted ↾ U or ( ) and we get an
X X U
⊆ O O O
LRS (U, ↾ U). Now, if we also have a collection of LRS’s (the standard models), we consider LRS’s,
X
O
(X, ), such that (X, ) is locally isomorphic to a standard model. This means that we can cover X by
X X
O O
opens and thatfor every openU X in this cover,there is a standardmodel (W, ) andanisomorphism
W
⊆ O
(U, ↾U)=(W, ), as LRS’s.
OX ∼ OW
Some Standard Models.
(1) Let U be an open ball in Rn or Cn, and let be the sheaf of germs of continuous functions on U
U
O
(this means, the sheaf such that for every open V U, (V) = the restrictions to V of the continuous
U
⊆ O
functions on U). If (X, ) is locally isomorphic to a standard, we get a topological manifold.
O
(2)LetU beanopenasin(1)andlet bethesheafofgermsofCk-functionsonU,with1 k . If
U
O ≤ ≤∞
(X, )islocallyisomorphictoastandard,wegetaCk-manifold(whenk = ,callthesesmooth manifolds).
O ∞
(3) Let U be an open ball in Rn andlet be the sheafof germs of real-valuedCω-functions on U (i.e.,
U
O
real analytic functions). If (X, ) is locally isomorphic to a standard, we get a real analytic manifold.
O
(4) Let U be an open ball in Cn and let be the sheaf of germs of complex-valued Cω-functions on U
U
O
(i.e., complex analytic functions). If (X, ) is locally isomorphic to a standard, we get a complex analytic
O
manifold.
(5) Consider an LRS as in (2), with k 2. For every x X, we have the tangent space, T , at x.
X,x
≥ ∈
Say we also have Q , a positive definite quadratic form on T , varying Ck as x varies. If (X, ) is locally
x X,x
O
isomorphic to a standard, we get a Riemannian manifold.
(6)Suppose W is open in Cn. Lookat some subset V W and assumethat V is defined as follows: For
⊆
any v V, there is an open ball B(v,ǫ)=B and there are some functions f ,...,f holomorphic on B , so
ǫ 1 p ǫ
∈
that
V B(v,ǫ)= (z ,...,z ) B f (z ,...,z )= =f (z ,...,z )=0 .
1 q ǫ 1 1 q p 1 q
∩ { ∈ | ··· }
The question is, what should be ?
V
O
We need only find out that what is (DX). We set = the sheaf of germs of holomorphic
OV∩Bǫ OV∩Bǫ
functions on B modulo the ideal (f ,...,f ), and then restrict to V. Such a pair (V, ) is a complex
ǫ 1 p V
O
analytic space chunk. An algebraic function on V is a ratio P/Q of polynomials with Q = 0 everywhere
6
on V. If we replace the term “holomorphic” everywhere in the above, we obtain a complex algebraic space
chunk.
Actually, the definition of a manifold requires that the underlying space is Hausdorff. The spaces that
we have defined in (1)–(6) above are only locally Hausdorff and are “generalizedmanifolds”.
Examples.
(1) Take W =Cq, pick some polynomials f ,...,f in C[Z ,...,Z ] and let V be cut out by
1 p 1 q
f = = f = 0; so, we can pick B(v,ǫ) = Cq. This shows that the example ( ) is a complex algebraic
1 p
··· †
variety (in fact, a chunk). This is what we call an affine variety.
10 CHAPTER 1. COMPLEX ALGEBRAIC VARIETIES; ELEMENTARY THEORY
Remark: (to be proved later) If V is a complex algebraic variety and V Cn, then V is affine.
⊆
This remark implies that a complex algebraic variety is locally just given by equations of type ( ).
†
(2) The manifolds of type (4) are among the complex analytic spaces (of (6)). Take W =B(v,ǫ) and no
equations for V, so that V B(v,ǫ)=B(v,ǫ).
∩
Say (X, ) is a complex algebraic variety. On a chunk, V W and a ball B(v,ǫ), we can replace
X
O ⊆
the algebraic functions heretofore defining by holomorphic functions. We get a complex analytic chunk
X
O
and thus, X gives us a special kind of complex analytic variety, denoted Xan, which is locally cut out by
polynomials but with holomorphic functions. We get a functor
X Xan
from complex algebraic varieties to complex analytic spaces. A complex space of the form Xan for some
complex algebraic variety, X, is called an algebraizable complex analytic space.
Take n+1 copies of Cn (Cn with either its sheaf of algebraic functions or holomorphic functions). Call
the j-copy U , where j =0,...,n. In U , we have coordinates
j j
Z(0),Z(1),...,Z(j),...,Z(n)
h j j j j i
(Here, as usual, the hat over an expressionmeans thatdthe correspondingitem is omitted.) For all i=j, we
6
have the open, U(i) U , namely the set ξ U (ith coord.)ξ(i) =0 . We are going to glue U(i) to U(j)
j ⊆ j { ∈ j | j 6 } j i
as follows: Define the map from U(i) to U(j) by
j i
Z(0) Z(i−1) Z(i+1) 1 Z(n)
Z(0) = j ,...,Z(i−1) = j ,...,Z(i+1) = j ,...,Z(j) = ,...,Z(n) = j ,
i Z(i) i Z(i) i Z(i) i Z(i) i Z(i)
j j j j j
withthecorrespondingmaponfunctions. Observethattheinverseoftheabovemapisobtainedbyreplacing
Z(i) with Z(j). However, to continue glueing, we need a consistency requirement. Here is the abstract
j i
requirement.
Proposition 1.1 (Glueing Lemma) Given a collection (U , ) of LRS’, suppose for all α,β, there exists
α OUα
an open Uβ U , with Uα =U , and say there exist isomorphisms of LRS’s,
α ⊆ α α α
ϕβ: (Uβ, ↾Uβ) (Uα, ↾Uα), satisfying
α α OUα α → β OUβ β
(0) ϕα =id, for all α,
α
(1) ϕβ =(ϕα)−1, for all α,β and
α β
(2) For all α,β,γ, we have ϕβ(Uβ Uγ)=Uα Uγ and
α α ∩ α β ∩ β
ϕγ =ϕγ ϕβ (glueing condition or cocycle condition).
α β ◦ α
Then, there exists an LRS (X, ) so that X is covered by opens, X , and there are isomomorphisms of
X α
O
LRS’s, ϕ : (U , ) (X , ↾X ), in such a way that
α α OUα → α OX α
(a) ϕ (Uβ)=X X ( X ) and
α α α∩ β ⊆ α
(b) ϕ ↾Uβ “is” the isomorphism ϕβ, i.e., ϕ ↾Uβ =ϕ ↾Uα ϕβ.
α α α α α β β ◦ α