complex algebraic geometry - fudan...

Introduction to Complex Algebraic Geometry

April 17, 2017

Chapter 1

An Overview

The complex algebraic geometry is the overlap of the complex geometry and algebraic geometry. It studiesthe systems of complex polynomial equations

8>><

>>:

f1

(x1

, . . . , xn

) = 0...

fm

(x1

, . . . , xn

) = 0,

(1.0.1)

where the fj

are polynomials in C[x1

, . . . xn

]. The solution set X of (1.0.1) is a topological space equippedwith both algebraic structure and analytic strucutre, i.e. it can be viewed as a complex algebraic variety andalso a complex analytic variety.

This course will talk about the elementary theory in this subject such as complex manifolds, Kahler geometry,projective varieties, sheaf theory and Hodge decomposition theorem.

3

4 CHAPTER 1. AN OVERVIEW

Chapter 2

Complex Geometry

2.1 Preliminary

Manifolds and Vector Bundles

A topological space M is a di↵erentiable manifold of dimension n if there exists an open cover {U↵

} andhomemorphisms

'↵

: U↵

! Rn, (2.1.1)

such that the transition function '↵�

:= '↵

� '�1

�

is smooth/di↵erentiable on '�

(U↵

\ U�

).

The maps (2.1.1) are called coordinate maps and the collection {(U↵

,'↵

)} is called a C1-atlas on M .

Examples

1. Any open subset of Rn.

2. R2/Z2.

3. Mobius band

We study the topology (cohomology groups) and geometry (bundles and metrics) of manifolds.

Definition 2.1.1. A reall (resp. complex) vector bundle of rank m over M is a topological space E equippedwith a map ⇡ : E ! M such that for an open cover {U

i

} of M , there exists local trivialisations

⌧i

: E|⇡

�1(Ui)

⇠= Ui

⇥ Rn

such that pr1

� ⌧i

= ⇡ and the transition functions

⌧j

� ⌧�1

i

: ⌧i

(E|⇡

�1(Ui\Uj)

) ! ⌧j

(E|⇡

�1(Ui\Uj)

)

are linear on each fiber.

A section of a vector bundle ⇡ : E ! M is a continuous map s : M ! E such that the composition ⇡ � s isthe identity.

Given a vector bundle E, we can define the dual bundle E⇤ = Hom(E,R) and its exterior power ^kE.

5

6 CHAPTER 2. COMPLEX GEOMETRY

Tangent Bundle

Definition 2.1.2. The tangent bundle TM

of M is described as follows: given an atlas {(U↵

,'↵

)} of M ,then T

M

is trivialized by Ui

⇥M and the transition functions (U↵

\ U�

)⇥ Rm ! (U↵

\ U�

)⇥ Rm are givenby

(u, v) ! (u,'↵�⇤(v)).

The points of TM

can be identified with equivalence classes of di↵erentiable maps � : [�✏, ✏] ! M for theequivalence relation

�1

⌘ �2

, �1

(0) = �2

(0), �01

(0) = �02

(0).

Another description

A tangent vector at a point x can be viewed as a derivation of the algebra of the real di↵erentiable functionson X supported at x, i.e. the linear maps : C1(X) ! R satifying the Leibniz’ rule at a point x 2 M .

Definition 2.1.3. A connected manifold M is orientable if the di↵eomorphisms '↵

� ��1

�

have positiveJacobian.

Examples

1. Compact orientable surfaces are called compact Riemann surfaces. They are classified by their genus.

2. Mobius strip is not orientable.

Exercise 2.1.4. Simply connected manifolds are orientable.

The cotangent bundle ⌦1

M

= T ⇤M

and ⌦k

M

:= ^k⌦1

M

. The sections of ⌦k

M

are called di↵erential k-forms onM . For a n-form ↵ on M , it can be integrated over M via coordinates maps. The following result followsfrom the Stoke’s formula in the Euclidean space.

Theorem 1 (Stoke’s formula: simple version). Let M be an orientable compact manifold of dimension n.Let ↵ be di↵erential n� 1 form on M . Then

RM

d↵ = 0

De-Rham and Betti cohomology

We denote by Ak(M) the space of sections of ⌦k(M). It is called the space of di↵erential k-forms on M .

Example. If M = Rn, Ak(M) = {PI

fI

(x)dxi1

^ . . . ^ dxik

}.

There is a natural exterior di↵erentiale operator

d : Ak�1(M) ! Ak(M)

satisfying d2 = 0. Locally, it is defined by

d(fI

(x)dxI

) =X

i

@fI

@xi

dxI

^ dxi

.

2.1. PRELIMINARY 7

Definition 2.1.5. We define the de-Rahm complex of M as

· · ·Ak�1(M)d�! Ak(M)

d�! · · · (2.1.2)

Its cohomology is the de-Rham cohomology of M , denoted by H⇤dR

(M,R).

Besides the De-Rham cohomology, we shall introduce the singular cohomology of a di↵erentiable manifold.Deine these as follows:

�k = {kX

i=0

ti

= 1, ti

� 0}

Ck

(M) =< f : �k ! M >

@k

: Ck

(M) ! Ck�1

(M)

@k

(�) =X

i

(�1)i�|[v0,...,vi,...,vn]

,

(2.1.3)

where [v0

, . . . , vi

, . . . , vn

] is the i-th face of �k

.

As @2 = 0, we obtain the so called singular complex (C⇤(M), @) of M .

Consider the dual complex (C⇤sing

(M), d) of (C⇤(M), @), one defines singular cohomology of M as the coho-mology group of (C⇤

sing

(M), d), denoted by H⇤sing

(M,Z). The following theorem is due to de Rham.

Theorem 2. Hk

dR

(M,R) ⇠= Hk

sing

(M,Z)⌦ R

Definition 2.1.6. We define the k-th Betti number

bk

= rank Hk

sing

(M,Z)

and the Euler characteristic of M as�(M) =

X

i�0

(�1)ibi

.

Important methods to compute cohomology groups: CW complexes, Mayer-Vietoris Sequences.

Interesting remark. Euler’s polyhedron formula: V �E + F = 2. This can be viewed as a toy application ofCW complexes.

Some Fact.

1. H0(M,Z) = ]{connected component}, H>n(M,Z) = 0.

2. Poincare dudality: if M is connected oriented and compact, the integration gives an isomorphism

H i(M,R)⌦Hn�i(M,R) ! R,

defined by (↵,�) !RM

↵ ^ �.

Recall the classical Euler formula (R+ V � E) = 2

There is another usual cohomology theory: Cech cohomolgy. It will be discussed in the later sections


Multivariable Holomorphic Functions

Let U be an open subset of Cn.

Definition 2.1.7. A function f : U ! C is said to be holomorphic if the di↵erential df is C-linear. Orequivalently, f is holomorphic if and only if near each point z

0

2 U , f admits an expansion as power series

f(z0

+ z) =X

I

aI

zI .

Theorem 3 (Cauchy’s Integral Formula: one variable). Let f be a smooth function on U and D a closeddisk contained in U . Then for every point z

0

2 D,

f(z0

) =1

2⇡i

Z

@D

f(z)

z � z0

dz +1

2⇡i

Z

D

@f

@w

dw ^ dw

w � z.

In particular, if f is homolomorphic, we have

f(z0

) =1

2⇡i

Z

@D

f(z)

z � z0

dz

Proof. This is based on Stokes’ formula. Consider the di↵erential form ⌘ = 1

2⇡i

f(w)dw

w�z

, we have

d⌘ =1

2⇡i

@f

@w

dw ^ dw

w � z.

If we use polar coordinates w = z + rei✓, we have

d⌘ = � 1

⇡

@f

@w(z + rei✓)dr ^ d✓,

which can be defined smoothly on D.

Let D✏

be the closed disc of raudis ✏ around z0

. The form ⌘ is smooth in D�D✏

. Applying Stokes’ theoremon D �D

✏

, we obtain1

2⇡i

Z

D�D✏

d⌘ =1

2⇡i

Z

@D

⌘ � 1

2⇡i

Z

@D✏

⌘

By taking limit ✏! 0, we obtain the assertion.

|

After repeatedly using one variable Cauchy’s formula, we can obtain the multi-variable version

Theorem 4. If f is holomorphic on U ✓ Cn and D a closed polydisk contained in U , then f(z) =( 1

2⇡i

)nR|⇠i�zi|=1,i=1...,n

f(⇠) d⇠1⇠1�z1

^ . . . d⇠n⇠n�zn

, where z = (z1

, . . . , zn

).

Applications. Maximaum Principle and Principle of Analytic Coninuation.

The following theorem is a preparation of the @-Poincare Lemma. Again, we only need to deal with the onevariable case.

2.2. COMPLEX MANIFOLDS 9

Theorem 5 (@-Lemma). Let f be a smooth function on an open set of C. Then locally on this open set,there exists a smooth function g such that

@g

@z= f

Proof. As this is a local question, we can assume that f(z) has compact support. Set

g(z) =1

2⇡i

Z

C

f(w)

w � zdw ^ dw := lim

✏!0

Z

C�D✏

f(w)

w � zdw ^ dw.

Then we can write

g(z) =1

2⇡i

Z

C

f(w)

w � zdw ^ dw

=1

2⇡i

Z

Cf(u+ z)

du ^ du

u

(2.1.4)

where u = w � z. Changing to polar coordinates u = rei✓, this integral becomes

g(z) = � 1

⇡

Z

Cf(z + rei✓)e�i✓dr ^ ✓,

which is a smooth function in z. It remains to show that g satisfies the equation. Note that

@g(z)

@z= � 1

⇡

Z

C

@f

@z(z + rei✓)dr ^ d✓

=1

2⇡i

Z

D

@f(w)

@w

dw ^ dw

w � z

= f(z)

by Cauchy integral formula.

|

2.2 Complex Manifolds

Definition 2.2.1. A complex manifold X is a di↵erentiable manifold admitting an open cover {U↵

} andcoordinate maps

'↵

: U↵

! Cn (2.2.1)

such that the transition function '↵�

= '↵

� '�1

�

is holomorphic on '�

(U↵

\ U�

) for all ↵,�. We say that{U

↵

,'↵

} is a complex structure of X.

Conventions: In this note, we will use UI

to denote the intersectionTi2I

Ui

, where I is an index set.

Examples

1. A one dimensional complex manifold is called a Riemann surface.

2. Cn is a (non-compact) complex manifold of dimension n with the natural coordinates.


3. The projective spacePn

C = Cn+1 � {0}/ ⇠

is a (compact) complex manifold of dimension n. The local coordinates are Ui

= {xi

6= 0} withcoordinate map

'i

(Z0

, . . . , Zn

) = (Z0

Zi

, . . . ,Zi

Zi

, . . .Zn

Zi

).

On 'j

(Ui

\ Uj

) ✓ Cn with coordinate (w1

, . . . , wn

), the transition functions

'ij

(w1

, . . . , wn

) = (w1

wj

, . . . ,wn

wj

)

are rational functions, which are obviously holomorphic.

4. Compact torus Cn/Z2n.

Exercise 2.2.2. Show that Pn

C is a compact oriented complex manifold and compute its cohomology.

Definition 2.2.3. A smooth complex vector bundle ⇡ : E ! M over a complex manifold M is said to beholomorphic if it is equipped with a holomorphic structure, i.e. there exist trivialisations

⌧i

: ⇡�1

i

(Ui

) ⇠= Ui

⇥ Cn

such that the transition matrices ⌧ij

= ⌧j

� ⌧�1

i

have holomorphic coe�cients.

Example (Holomorphic Tangent bundle) Given a complex chart {�i

: Ui

⇠= Cn}, we can define the holomor-phic tangent bundle T

M

:= T 1,0

M

on M as the union of Ui

⇥Cn, glued by identifying (Uij

)⇥Cn and Uji

⇥Cn

via(u, v) ! (u,�

ij⇤(v)).

Here the holomorphic Jacobian matrix �ij⇤ is the matrix with holomorphic coe�cients

@�

kij

@zI.

Let us describe the relation between holomorphic tangent bundle and real tangent bundle via their fibers.The tangent space of a point p 2 M can be described as below

• Tp,M

is the usual real tangent space of dimension 2n. In a neighbouhood of p, if we write zk

= xk

+ iyk

,then

Tp

(M) = R{ @

@xk

,@

@yk

}

• The complexified tangent space TCp,M

= Tp,M

⌦ C is the space of C-lienar derivations and we have

TCp,M

= C{ @

@zk

,@

@zk

}

• T 1,0

p,M

= C{ @

@zk} ✓ TC

p,M

is called the space of holomorphic tangent vectors at p 2 M . Moreover, wehave

TCp,M

= T 1,0

p,M

� T 0,1

p,M

(2.2.2)

where T 0,1

p,M

= C{ @

@zk} and T 1,0

p,M

= T 0,1

p,M

.


Let f : M ! N be a smooth map. Then it induces a linear map

f⇤ : Tp,C(M) ! Tf(p),C(N)

We say f is a holomorphic map if f⇤(T1,0

p

(M)) ✓ T 1,0

f(p)

(N) for all p 2 M , or equivalently, for all charts

{(U↵

,'↵

)} on M and {(V�

,��

)} on N , ��

� f � '�1

↵

is holomorphic where it is defined.

Definition 2.2.4. We can define so called complex submanifolds of M , which says that N is a complexsubmanifold of M if there exists a holomorphic closed embedding N ,! M .

Definition 2.2.5. A complex manifold M is projective if M is a closed complex submanifold of PN

C for someN .

Almost Complex Structure and Frobenius Theorem

Let M be a smooth di↵erentiable manifold.

Definition 2.2.6 (Almost complex structure). An almost complex structure on M is an edomorphism

J : TM

! TM

such that J2 = �id.

An easy fact is that if M is complex manifold, then there is a natural almost complex structure on M . Thisyields

Definition 2.2.7. We say that J is integrable on M if there exists a complex structure on M which inducesJ .

With the almost complex structure, we have a decomposition of the complexified tangent bundle TM

⌦ C

TM

⌦ C = T 1,0

M

� T 0,1

M

where T 0,1

M

and T 1,0

M

are eigenspace of J . In particular, when M is a complex manifold and J is the induced

almost complex structure, then T 1,0

M

is equal to the holomorphic tangent bundle. Moreover, there is a bracket[ , ] on T

M

⌦ C defined by8X,Y 2 T

M

⌦ C, [X,Y ] = X � Y � Y �X.

Locally (as a manifold with real coordinate (x1

, . . . , xn

)), the bracket [�, ] can be expressed as

[ ,�] =X

i

(�( i

)� (�i

))@

@xi

, where =P i

@

@xiand � =

P�i

@

@xi. (2.2.3)

Question. When is the almost complex structure integrable?

Theorem 6 (Newlander-Nirenberg). An almost complex structure J is integrable if and only if [T 0,1

M

, T 0,1

M

] ⇢T 0,1

M

.

Proof. Let us divide the proof into several steps.


1. First, we start with the tranditional Frobenius theorem, which gives a criteria for the integrabilityof real distributions.

Definition 2.2.8. A distribution E on M is a C1-vector subbundle of TM

. We say E is integrable ifM is covered by open subsets U

i

such that there exists a C1-map

�i

: Ui

! Rn�rankE ,

satisfying 8p 2 Ui

, the vector subspace Ep

✓ Tp,M

is equal to ker d�p

.

Theorem 7 (Frobenius Theorem). A distribution E is integrable if and only if [E,E] ✓ E.

Sketch of the Proof.

()) If E is integrable, since the sections of E are exactly vector fields which annihilate the coordinatefunctions �

k

= xk

� �i

, it follows from (2.2.3) that [E,E] ✓ E.

(() By induction on rank E. When rank E = 1, then E is generated by a single vector field. Theassertion follows from a (flow-box) theorem of Arnold.

Suppose it holds when rank E = m � 1. First, one can find open subsets U such that there exists anon-zero section � of E over U and a submersive map

� : U ! Rn�1

whose fibers are the trajectories of �. Moreover, we can assume U⇠�! V ⇥ R and � = @

@t

by flow-boxtheorem. The key fact is that E is integrable if only only if E is induced by an integrable distributionon V (proof omitted).

2. (Holomorphic Version) (Foliation)

Theorem 8. Let M be a complex manifold and let E be a holomorphic distribution, i.e. E ✓ T 1

M

.Then E is integrable in the holomorphic sense if and only if [E,E] ✓ E.

Idea:

• Reduce to the case of real Frobenius theorem. Note that [E,E] ✓ E implies that <E (as a realdistribution Re(E) � JRe(E)) is integrable, which means that M is covered by open subsets Usuch that

�U

: U ! V ✓ R2(n�k)

with <E = ker(�U,⇤ : TU

! TR2(n�k)), where k = rank E.

• It su�ces to show that there exists a complex structure on V such that � is holomorphic. There isan induced almost complex structure on the tangent bundle T

V

= TU

/<E as <E is stable underthe complex structure J 2 End(T

M

). Moreover, this almost complex structure is independent ofthe choice of the points on the fiber of �. Now we can take a complex submanifold of U transverseto the fibers of �. Via �

U

, this submanifold is locally isomorphic to V and compatible with thealmost complex structure. Thus the almost complex structure on V is integrable.

3. (M is real analytic)


(1) If there is an almost complex structure J satisfying the assumption, we can assume that M = U isan open subset of R2n and J is a real analytic map with values in End(R2n).

Next, one can take an open subset UC ✓ C2n, which is a neighbourhood of U in C2n and J extendsto UC as a End(C2n)-valued holomorphic function. Consider the holomorphic distribution

EC ✓ T 0,1

UC

to be the eigenspace of J with respect to the eigenvalue �i. Then the restriction of EC to U is justT 0,1

U

.

(2) It is not di�cult to check that the holomorphic distribution EC satisfies the condition [EC, EC] ✓ ECand thus it is integrable. So it gives arise to a holomorphic submersion

�C : UC ! Cn,

whose fiber are the integral holomorphic submanifolds of the distribution EC.

(3) Finally, we have to show that the restriction of �C to U is a local di↵eomorphism and the inducedcomplex structure coincides with the almost complex structure J . The first claim follows from thefact �C,⇤|U is an isomorphism from the construction. The second one is pretty clear as everythingis canonical.

|

Remark 1. In general, both the existence and integrability of almost complex structure are very di�cult.

• (Borel-Serre) The only spheres which admits almost complex structures are S2 and S6.

• After 60 years, Atiyah recently claimed a “proof” for the non-existence of complex structure on S6.

Examples

1. Riemann Surface.

Theorem 9. Riemann surfaces are one dimensional complex manifolds.

Proof. Let ⌃ be a compact oriented smooth surface. Then ⌃ comes with an almost complex structure.The existence of almost complex structure actually follows from the existence of a Riemannian metricon ⌃.

Recall that a Riemann metric g on ⌃ means that it assigns an inner product on the tangent space Tp,⌃

at each point, which varies smoothly, i.e. the function

pg�! (X(p), Y (p))

is smooth for any two tangent vector fields X,Y . If g is a metric on Tp,⌃

, then we can define

Jp

: Tp,⌃

! Tp,⌃

be the unique element of SO(Tp,⌃

, gp

) satisfying J2

p

= �1 and (u, Ip

u) forms a positively oriented basisof T

p,⌃

for every u 6= 0. At last, J is integrable because the condition in Newlander-Nirenberg Theoremis trivially satisfied.

Conversely, if Jp

is such a complex operator on Tp,⌃

, then there exists a metric gp

on Tp,⌃

which satisfiesthat condition g(Iu, Iv) = g(u, v) for u, v 2 T

p,⌃

. |


2. Complex Torus/ Abelian Variety. The quotient Cn/Z2n is a complex torus, which is a compact complexmanifold of dimension n.

Di↵erential operators

As the cotangent bundle ⌦M,C admits a decomposition

⌦M,C = ⌦1,0

M

� ⌦0,1

M

.

Then the complexified bundle of di↵erential k-forms ⌦k

MC =V

k ⌦M,C decomposes into

X

p+q=k

⌦p,q

M

,

where ⌦p,q

M,C = ^p⌦1,0

M,C ⌦ ^q⌦0,1

M,C.

Denote by A•M

the space of section of the (complex) vector bundles above. There are three operators d, @, @on A•

M,C.

d : Ak(M) ! Ak+1(M);

@ : Ap,q(M) ! Ap+1,q(M);

@ : Ap,q(M) ! Ap,q+1(M);

satisfyingd = @ + @, d2 = @2 = @2 = 0.

They satisfy the Leibniz’ rule@(↵ ^ �) = @↵ ^ � + (�1)deg↵↵ ^ @�

(Similarly for @) and an identity @↵ = @↵.

Locally, if ↵ =P↵I,J

dzI

^ dzJ

with |I| = p, |J | = q, then

@(↵) = @↵I,J

^ dzI

^ dzJ

, @(↵) = @↵I,J

^ dzI

^ dzJ

.

We have the following local exactness result of the di↵erential operator @.

Theorem 10 (Poincare Lemma). Let ↵ be a smooth form of type (p, q). If @↵ = 0, then locally there existsa smooth form � of type (p, q � 1) such that ↵ = @�.

Proof. Write ↵ =P↵I,J

dzI

^ dzJ

. Then

@↵ =X

I,J

@↵I,J

^ dzI

^ dzJ

.

So if @↵ = 0, then the (0, q)-form

↵I

=X

↵I,J

dzJ

is @-closed. If ↵I

= @�I

, then

↵ = (�1)p@(X

dzI

^ �I

)

2.3. KAHLER METRIC AND KAHLER GEOMETRY 15

Therefore, it su�ces to prove the case when p = 0. When p = 0, we can prove the assertion by induction onthe largest integer k 2 J with ↵

J

6= 0. If k = q, then ↵ is of the form

↵ = f(z)dz1

^ dz2

. . . dzq

and the assertion is equivalent to the Poincare lemma for holomorphic functions.

Suppose it holds for k � q + 1. We consider the di↵erential form

↵ = ↵1

+ ↵2

^ dzk+1

where the index of coordinates occuring in ↵1

and ↵2

are strictly less than k. As @↵ = 0 and hence thecoe�cients of ↵

2

with index � k + 1 are all holomorphic, one can easily find � and � such that

↵2

^ dzk+1

= @� + �,

and the coordinates in � have index k. Now, as @↵ = @(↵1

+ �) = 0, then ↵1

+ � = @(!) by assumption.

Dolbeault Complex of holomorphic vector bundle

Let X be a complex manifold.

Definition 2.2.9. E is a holomorphic bundle over X if the transition functions are holomorphic. We denoteby A0,q(E) the space of smooth sections of the bundle ⌦0,q

X

⌦ E.

The usual @-operator induced an operator on E

@E

: Ak(E) ! Ak+1(E)

So there is a natural associated Dolbeault complex of E

· · ·A0,q�1(E)¯

@E��! A0,q(E)¯

@E��! A0,q+1(E) · · · (2.2.4)

The cohomology of (2.2.4) is called the Dolbeault cohomology of E. The local exactness result also holds for@E

.

2.3 Kahler Metric and Kahler geometry

Definition 2.3.1 (Hermitian metric on complex manifolds). A Hermitian metric h on a complex manifoldM is a collection of Hermitian metric h

x

on each tangent space TM,x

. The Hermitian metric h defines aRiemannian metric via <h and a (real) Kahler 2-form ! = �=h. ! is alternating and of type (1, 1).

Then

(1) h(Ju, Jv) = h(u, v), !(Ju, Jv) = !(u, v)

(2) h(u, v) = !(u, Jv)� i!(u, Jv).


Conversely, given a (1, 1)-form !, we say ! is positive if

h = w(u, Jv)� iw(u, Jv)

is positive definite.

Definition 2.3.2. A Kahler metric on M is a Hermitian metric whose Kahler form is closed and we say Mis a Kahler manifold if it admits a Kahler metric.

Example 2.3.3. 1. M = Cn and the tangent bundle TM

= Cn ⇥ Cn. The standard Hermitian metric

hz

(u, v) = u · v

Its corresponding Kahler form is

! =i

2

X

k

dzk

^ dzk

which is obivously closed.

2. The complex torus Cn/⇤ inherits a flat Kahler metric induced by the Kahler metric on Cn.

Proposition 1. Let M be a complex manifold.

1. The volume form associated to a Hermitian metric h on M is !

n

n!

.

2. If M is compact and Kahler, then for every integer k between 1 and n, the closed form !k is not exact.

3. If M is Kahler, the complex submanifolds of M are Kahler and they can not be a boundary in M .

4. Let h be a Kahler metric on M . For every point p 2 M , there exists holomorphic coordinates z1

, . . . , zn

centered at p such that hij

= h( @

@zi, @zj¯

@zj) is I

n

+O(P

|zi

|2).

Proof. (1) Recall that the volume form is an everythere non-zero section of ⌦2n

M

and has norm 1 on ⌦p,M

.For x 2 M , we can take an orthogonal basis of T

x,M

over C, named {e1

, . . . , en

}. Then {ei

, Jei

} is a realbasis of T

x,M

. The volume form of (M,h) at x is the unique form which has value 1 on

e1

^ Je1

^ . . . ^ Jen

.

Now, if we write the coordinates at x as zk

= xk

+ iyk

and hence ! = i

2

Pdz

k

^ dzk

, then

!n

n!= (

i

2)n

Ydz

k

dzk

= dx1

^ dy1

^ dx2

. . . . . . ^ dyn

(2) If !k = d� is exact, then !n = d(!n�k ^ �) is exact. However, it contradicts to the Stokes theorem.

(3) If N ,! M is a complex submanifold, then the induced Kahler metric on N is Kahler. Moreover, ifN = @(X) with X ,! M . By Stoke’s theorem, we have

Z

N

(!M

|N

)dimN =

Z

X

d(!M

|X

)dimN = 0,

as !M

is closed.


(4) By choosing appropriate coordinates, we can let (hij

) = In

at p. So it can be written as

h =X

dzi

dzi

+X

✏ij

dzi

dzj

+O(|z|2),

where ✏i,j

are linear functions of zi

and zi

. Since (✏ij

) is Hermitian, then

✏ij

= ✏holij

+ ✏holji

Note that

! =i

2(X

dzi

^ dzi

+X

✏ij

dzi

^ dzj

+O(|z|2))

is closed at p, we get X✏holij

dzi

^ dzj

is @-closed at p and hence closed everywhere. So we have

@✏holij

@zk

=@✏hol

kj

@zi

(2.3.1)

This implies that there exists holomorphic functions �j

(z1

, . . . , zn

) such that

✏holij

=@�

j

@zi

.

By changing the coordinate z0i

= zi

+ �i

(z), one can check

! =i

2

Xdz0

k

^ dz0k

+O(|z0|2).

|

Chern form of holomorphic line bundles

Let L be a holomorphic line bundle on a complex manifold M . Then locally, L admits a holomorphictrivialisation

L|Uj

⇠= Uj

⇥ C

where {Uj

} is an open cover of M . Let h be a Hermitian metric on L and hj

= h|Uj . Then the 2-form

!j

=1

2⇡i@@ log h

j

defines a 2-from on ! on M as !i

and !j

concide on the intersection Ui

\ Uj

.

Definition 2.3.4. The 2-form defined as above is called theChern form of L.

An important fact is that if the line bundle L satisfies some positive condition, then the Chern form ! ispositive. This will give arise to a Kahler metric on the complex manifold.


More Examples (I): Fubini-Study Metric

On the projective spacce Pn

C, there is a natural holomorphic line bundle OPn(�1) whose fiber at [x0

, . . . , xn

]is the line `

(x0,...,xn).

Construction of Hermitian metric on Pn.

Let h be the standard Hermitian metric on Cn+1. As OPn(�1) ✓ Pn ⇥Cn, then it gives arise to a Hermitianmetric h on OPn(�1) by restriction. Let h⇤ be the corresponding metric on OPn(1) and let ! be its associatedChern form. Locally, !

j

:= !|Uj is of the form

1

2⇡i@@ log h⇤(�⇤

j

)

where �j

(z1

, . . . , . . . , zn

) = (z1

, . . . , 1, . . . , zn

) is the section of OUj (�1). From the definition, we have

h(�j

) = 1 +X

k 6=j

|zk

|2; and !j

=1

2⇡i@@ log(

1

1 +Pk 6=j

|zk

|2 ).

Theorem 11. The Hermitian metric associated to ! is a Kahler metric.

Proof. It su�ces to show that ! is positive. As this is a local property, we can check on each !j

.

Due to a direct computation, we have

!j

=i

2⇡

(1 +Pk 6=j

|zk

|2)P

dzk

^ dzk

� (P

zk

dzk

) ^ (P

zk

dzk

)

(1 +Pk 6=j

|zk

|2)2 .

At the point 0, we have !j

= i

2⇡

Pk 6=j

dzk

^ dzk

is positive. As ! is invariant under SU(n+ 1) and SU(n+ 1)

acts transitively on Pn, it follows that ! is everywhere positive.

|

Definition 2.3.5. Let E ! M be a holomorphic vector bundle of rank r + 1 over M . The associatedprojective bundle ⇡ : P(E) ! M is defined to be the quotient of E minus the zero section by the naturalaction of C⇤.

Clearly, P(E) is also a complex manifold.

Corollary.

(a) Projective manifolds are Kahler manifolds.

(b) The projective bundles over a compact Kahler manifold are Kahler.

(c) The blow up of a Kahler manifolds along a compact complex submanifolds are Kahler.


Proof. (a) is obvious. Here we only give a proof of (b). (c) will be left to readers.

On P(E), there is a relative tautological line bundle OP(E)

(�1). The Hermitian metric on E induces a metricon the dual line bundle OP(E)

(1). Its Chern form !E

is potive on each fiber of ⇡. As M is Kahler, it admitsa positive Kahler form !

M

.

Note that M is compact, then for su�ciently large �, the closed form

! = !E

+ �⇡⇤!M

is positive on P(E). Hence ! defines a positive Chern form of P(E). This proves the assertion. |

More Examples (II): Non Kahler manifolds

1. Hopf surfaces

Definition 2.3.6. A Hopf surface S is a compact complex surface obtained as a quotient of the complexvector space (with zero deleted) C2\{0} by a free action of a discrete group �.

As a simple example, we take � = {�n|n 2 Z} for some 0 < � < 1. Clearly, S = �\C2 � {0} is acomplex surface. Moreover, as C2 � {0} ⇠= R+ ⇥ S3 and � acts on R+-factor. So

S ⇠= S1 ⇥ S3

is compact. The Betti numbers of S are

b0

(S) = b1

(S) = b3

(S) = b4

(S) = 1, b2

= 0.

So S is not Kahler as its first Betti number is odd.

2. More generally, Kodaira and Siu showed that a compact complex surface has a Kahler metric if andonly if its first Betti number is even. (Topology determines the existence of Kahler structure.)

3. For higher dimensional complex manifolds, Hironaka’s example shows that this fails in dimensions atleast 3.

Chapter 3

Algebraic Varieties

3.1 A�ne and Projective Varieties

Definition 3.1.1. An a�ne algebraic closed subset X in Cn is the vanishing locus of a set of complexpolynomials, i.e.

X = V (f1

, . . . , fk

) := {p 2 Cn| fi

(p) = 0}.

Conversely, given X ✓ Cn, we define

I (X) = {f 2 C[x1

, . . . , xn

]| f(p) = 0, 8 p 2 X},

which is an ideal of C[x1

, . . . , xn

].

e.g. When X = (a1

, . . . , an

) 2 Cn is a point, then I (X) = (x1

� a1

, x2

� a2

, . . . , xn

� an

) is a maximal idealof C[x

1

, . . . , xn

]. Conversely, we have

Theorem 12 (Hilbert’s Weak Nullstellensatz). The maximal ideals of k[x1

, ..., xn

] are precisely those idealsof the form (x

1

� a1

, ..., xn

� an

), where ai

2 k.

Definition 3.1.2. The a�ne coordinate ring A(X) = C[x1

, . . . , xn

]/I (X).

If X is algebraic, there is a Zariski topology on X by defining the closed subsetes as V (I) for some subsetsI ✓ A(X). We say X is an algebraic variety if X is irreducible. This is equivalent to I (X) is a prime idea.

Proposition 2. (1) If I1

✓ I2

, then V (I1

) ◆ V (I2

).

(2) If X1

✓ X2

, then I (X1

) ◆ I (X2

).

(3) I (X1

[X2

) = I (X1

) \ I (X2

).

(4) 8 Y ✓ Cn, V (I (Y )) = Y .

(5) (Hilbert’s Nullstellensatz) If I is an ideal, thenpI = I (V (I)).

This estabilishes a correspondence

{prime ideals of A(X)} , {irreducible varieties of X}

21

22 CHAPTER 3. ALGEBRAIC VARIETIES

Remark 2. There is a purely algebraic definition of the dimension of X, called the Krull dimension ofX. For complex algebraic varieties, it is compatible with the classical definition of dimension in Euclideantopology.

Definition 3.1.3. Let f : X ! C be a function. It is regular at a point p if there is an open neighborhoodU such that f is a rational function on U . It is said to be a regular function if it is regular all points of U .We denote by O(X) the ring of all regular functions on X and O

P

the ring of germs of regular functions onX near a point P 2 X.

A morphism ' between two a�ne varieties X and Y is a continuous map, such that for any regular functionf on Y , the composition f � ' is regular on X.

Fact. OP

= O(X)mP , where m

P

is the corresponding maximal ideal.

For any morphism ' : X ! Y , it induces a ring homorphism

'⇤ : O(Y ) ! O(X)

via composition.

Theorem 13. Let Y be an a�ne variety. Then O(Y ) = A(Y ) and there is a bijection

Hom(X,Y )⇠�! Hom(A(Y ),O(X)).

Proof. There is an obvious injective map A(Y ) ! O(Y ). Moreover, we have O(Y ) = \P2Y OP

. Note that

OP

= A(Y )mP ,

it follows that A(Y ) = O(Y ).

For the second assertion, let us first assume that X is a�ne and hence A(X) = O(X). Then one canconstruct a map from Hom(A(Y ), A(X)) to Hom(X,Y ) as below: 8f : A(Y ) ! A(X), then f�1(m

p

) is amaximal ideal in A(Y ), denote by m

f

](p)

. This gives a map from f ] : X ! Y . Then it is easy to check this is

continuous in Zariski topology and f ] is regular from the definition. In general, one can take an a�ne opencover of X and apply the results above. |

* Spectrum of a ring

More generally, let A be a commutative ring. The spectrum of A, denoted by Spec(A), is the set of primeideals of A, It comes with the Zariski topology, for which the closed sets are

V (I) = {P 2 Spec(A)| I ✓ P}.

The open subsets of Spec(A) are constructed from Spec(Af

), where f 2 A.

We can also associate a ring to Spec(A) as below: O(Spec(A)) is the set of functions

Spec(A) !a

p2Spec(A)

Ap

,

where s(p) 2 Ap

. The pair (Spec(A),O) is a ringed space.

3.2. SMOOTH VARIETIES 23

Projective variety: Construction of Proj

Definition 3.1.4. A projective complex variety X is the vanishing locus of a set I of homogenous polynomialsin a projective space Pn. The homogenous coordinate ring S(X) := C[x

0

, . . . , xn

]/V (I) is a graded ring.

Locally, X is covered by a�ne varieties Xi

whose coordinate ring is (S(X)(xi)

)0

, i.e. the 0-graded piece ofthe localization of S(X) of x

i

.

Similarly, one can define a natural topology on X by taking the closed subsets as V (I) for homogenous idealsI ✓ S(X). It concides with the induced Zariski topology on a�ne varieties.

Proj construction. A Z-graded ring is a ring

S =M

n

Sn

,

where multiplication respects the grading, i.e. sends Sm

⇥Sn

to Sm+n

. Clearly S0

is a subring, each Sn

is anS0

-module, and S is a S0

-algebra. An ideal I of S is a homogeneous ideal if it is generated by homogeneouselements.

Write S+

:=Ld>0

Sd

. If S+

is a finitely generated ideal, We say S is finitely generated graded ring over S0

.

The Proj(S) is the collection of homogenous primes ideals of S not containing the irrelevant ideal S+

. Theclosed subsets on Proj(S) are the projective vanishing set V (I), where I ✓ S+. We call this the Zariskitopology on Proj(S).

To give the ringed space structure, we take the projective distinguished open set D(f) = Proj(S)\V (f) (theprojective distinguished open set) be the complement of V (f). One can show that D(f) is isomorphic toSpec((S

f

)0

). Hence it gives arise a natural ringed space structure on Proj(S).

3.2 Smooth Varieties

Definition 3.2.1. The Zariski cotangent space of X at a point P is mP

/m2

P

. We say X is nonsingular atP if dimm

P

/m2

P

= dimX.

Remark 3. One can define derivation of an k-algebra R as the C-linear map R ! R satisfying certainproperties such as the Leibnitz rule.

Obviously, Sing(X) is a Zariski closed subset of X.

In di↵erential geometry, the tangent space at a point is defined as the vector space of derivations at thatpoint. A derivation at a point P of a manifold is an operation that takes in functions f near P (i.e. elementsof O

P

), and outputs elements f 0, which satisfies the Leibnitz rule

(fg)0 = f 0g + g0f

A derivation is the same as a map mP

! R (by restriction), where mP

is the maximal ideal of OP

. (Con-versely, it extends to a map O

P

! R via the OP

! mP

defined by f � f(P )) But m2

P

maps to 0, as iff(P ) = g(P ) = 0, then

(fg)0(P ) = f 0(P )g(P ) + g0(P )f(P ) = 0.


Thus a derivation induces a map mP

/m2

P

! R, i.e. an element of (mP

/m2

P

)_.

Examples.

1. When X = V (f1

, . . . , fm

) is an a�ne variety of dimension n. Then X is smooth if and only if theJacobian matrix Jac has rank n.

2. Let X be a smooth hypersurface of degree d in Pn

C, i.e. X is the zero locus of an homogenous equation

f(x0

, . . . , xn

) of degree d. Then X is smooth if and only if ( @f@x0

, . . . , @f

@xn) is everywhere non degenerate.

3. X = {y2 = x2(x+ 1)}. At P = (0, 0), we have

OP

= (C[x, y]/(y2 = x2(x+ 1)))(x,y)

.

Its Zariski cotangent space is mP

/m2

P

⇠= C2. It is singular at (0, 0).

As complex manifolds

Recall that a complex manifold X is projective if there exists a closed embedding X ! Pn

C.

Theorem 14. A smooth projective complex variety is a projective manifolds.

This is due to the inverse function theorem and implicit function theorem.

Theorem 15 (Inverse Function Theorem). Let U and V be open subsets in Cn with 0 2 U . Let f : U ! Cn

be a holomorphic map whose jacobian is non-singular at the origin. Then f is one-to-one in a neighbourhoodof the original and the inverse is holomorphic near f(0).

Theorem 16 (Implicit Function Theorem). Write On

as the collection of holomorphic functions on Cn.Given f

1

, . . . , fk

2 On

with

det(J (f1

, . . . , fk

)(0)) 6= 0,

there exists functions !1

, . . . ,!k

2 OCn�k such that in a neighborhood of 0 in Cn,

f1

(z) = . . . = fk

(z) = 0 , zi

= !i

(zk+1

, . . . , zn

).

Corollary. A smooth complex algebraic variety is a smooth complex manifold.

An amazing fact is that the converse is also true.

Theorem 17 (GAGA). A smooth projective complex manifold is a smooth projective complex variety.

3.3 Examples

A toy example. Let A2 be the a�ne plane. We can define a variety Bl(0,0)

A2 of A2 ⇥ P1 as follows. If thecoordinates on A2 are x, y, and the projective coordinates on P1 are z

0

, z1

, this variety is cut out in A2 ⇥ P1

by the single equationxz

0

= yz1

.

3.3. EXAMPLES 25

One can interpret Bl(0,0)

A2 as follows. The P1 parametrizes lines passing through the origin. The blow-upcorresponds to ordered pairs of (P , line `) such that (0, 0), P 2 `. The natural projection

' : Bl(0,0)

A2 ! A2

is a birational morphism. The fiber '�1(0) is called the exceptional divisor.

Blow up. Let X be a smooth variety and Y ✓ X a smooth closed subvariety defined equations

f1

= f2

= . . . fk

= 0

Then the blow up of X along Y is defined as

BlY

(X) = {(z, Z) 2 X ⇥ Pk�1| Zi

fi

= Zj

fj

}.

The natural projection BlY

(X) ! X is an isomorphism outside Y .

Chapter 4

Sheaves and cohomology

4.1 Sheaves

Let X be a topological space.

Definition 4.1.1. A presheaf F of abelian groups over X consisting of the data

1. for every open subset U ✓ X, F(U) is an abelian group;

2. for each inclusion V ,! U , a morphism ⇢UV

: F(U) ! F(V )

subject to the conditions

• F(;) = 0, ⇢UU

= id

• if W ✓ V ✓ U , then ⇢UW

= ⇢VW

� ⇢UV

.

The stalk Fx

of a sheaf F over X at a point x is equal to lim�!

F(U).

Definition 4.1.2. A morphism between two presheaves ' : F ! G is a collection of morphisms '(U) :F(U) ! G(U) for each open subset U such that if V ✓ U , the diagram

F(U)'(U)

//

⇢UV

✏✏

G(U)

⇢

0UV✏✏

F(V )'(V )

// G(V )

is commutative.

Conventions: for s 2 F(U) and V ✓ U , we will write s|V

:= ⇢UV

(s) as the restrition of s to V . We writesx

as the germ of s at x.

Definition 4.1.3. A presheaf F is a sheaf if it satisfies two more axioms

27

28 CHAPTER 4. SHEAVES AND COHOMOLOGY

• (Identity axiom) If {Ui

}i2I is an open cover of U and if s 2 F(U), and s|

Ui = 0 for all i 2 I, thens = 0.

• (Gluability axiom) If {Ui

}i2I is an open cover of U and we have s

i

2 F(Ui

) such that si

|Uij = s

j

|Uij ,

then there exists s 2 F(U) such that si

= s|Ui .

The two axioms can be interpreted as the exactness of the equalizer exact sequence

! F(U) !Y

F(Ui

) !! F(Uij

)

Proposition 3. For every presheaf F , there exists a unique sheaf F+ over X with the property: there existsa morphism of preshaves

✓ : F ! F+

such that for any sheaf G, and any morphism : F ! G, there is a unique morphism ' : F+ ! G with = ' � ✓.

Proof. The sheaf F+ is constructed as below:

F+(U) = {s : U ![

x2UFx

| s(x) 2 Fx

, and s|V

2 F(V ) for some V 3 x}

|

In modern language, a sheaf is a functor from the category of open sets, where the objects are the open setsand the morphisms are inclusions, to the category of sets. The sheafification defines a functor

Category of presheaves over Xsheafification��! Category sheaves over X.

Examples.

1. (Constant sheaf)

Let X be a topological space and S a set. Define Spre

(U) = S for all open sets U. You will readilyverify that S

pre

(U) forms a presheaf (with restriction maps the identity). This is called the constantpresheaf associated to S. This isn’t (in general) a sheaf. But even if we patch the definition by settingSpre

(;) = e, if S has more than one element, and X is the two-point space with the discrete topology,Spre

fails gluability. We denote by S as its sheafification, called the constant sheaf associated to S.

2. (Skycraper sheaf)

Suppose X is a topological space, with a point p 2 X,and S is a set. Let ip

: p ! X be the inclusion.Then i⇤

p

S defined by

i⇤p

(U) =

(S; if p 2 U ;

{e}; if p /2 U.(4.1.1)

forms a sheaf. Here {e} is any one-element set. This is called a skyscraper sheaf.

4.1. SHEAVES 29

3. (Sheaves of sections of vector bundles) Let X be a smooth manifold. We denote by C1(X) the sheafof smooth functions on X, where C1

X

(U) = {smooth functions on X}.If X is a complex manifold, we write O

X

for the sheaf of holomorphic functions, where OX

(U) ={ring of holomorphic functions on U}.If E ! X is a vector bundle, the presheaf E

U 7! {continous sections of E|U

! U}

is a sheaf.

4. Pushforward and inverse image

Suppose ⇡ : X ! Y is a continuous map of topological spaces and F is a sheaf over X. Then one candefine the pushforward sheaf ⇡⇤F over Y as

⇡⇤F(U) = F(⇡�1(U)).

If G is a sheaf on Y , then we define the inverse image sheaf ⇡�1G over X as the sheafification of thepresheaf

U 7! lim�!⇡(U)⇢V

G(V ).

One can check that ⇡�1Gx

= G⇡(x)

.

Adjointness. Let ⇡ : X ! Y be a continuous map. If F is a sheaf over X and G is a sheaf over Y ,there is a bijection

Hom(⇡�1G,F) $ Hom(G,⇡⇤F).

5. Ideal sheaf.

Let i : Y ,! X be a closed subvariety. Then we define the ideal sheaf IY

as

IY

(U) = {f 2 OX

(U)| f(Y ) = 0}

Then there is an exact sequence0 ! I

Y

! OX

! i⇤OY

! 0.

6. Locally free sheaves, Coherent sheaves.

LetA be a sheaf of rings overX, i.e. A(U) is a ring and the restriction map ⇢UV

is a ring homomorphism.e.g. the structure sheaf O

X

is a sheaf of rings.

Definition 4.1.4. A sheaf F of A-modules over X is a sheaf F such that F(U) is an A(U)-modulecompatible with the restriction maps. We say that F is locally free if F

x

is a free Ax

module for allx 2 X. e.g. the sheaf E (of smooth sections) associated to a vector bundle is a sheaf of C1

X

-modulesand it is locally free.

Definition 4.1.5. A quasi-coherent sheaf on a ringed space (X,OX

) is a sheaf F of OX

-modules suchthat 8x 2 X, there is an open neighborhood U 3 x with

O�I

X

|U

! O�J

X

|U

! F|U

! 0

for some sets I and J . It is coherent if it is of finite type over OX

and for any open set U ✓ X, andany morphism On

X

|U

'�! F|U

, the kernel of ' is of finite type.


Example 4.1.6. Let X = Spec(A), where A is a Noetherian ring. For any A-module M , we can

associate a quasi-coherent sheaf fM byfM(D(f)) = M

f

.

Conversely, all quasi-coherent sheaf on X is of the form fM . Moreover, fM is coherent if and only if Mis a finitely generated A-module.

4.2 Resolutions

Definition 4.2.1. Let ' : F ! G be a morphism between two sheaves. The kernel (image) of ' are definedas the sheaf associated to the preshaf

U 7! ker('(U))(resp. image('(U))).

written as ker(') and im(').

We say ' is injective if ker(') = 0 and surjective if im(') = G. This also allows us to define the exactsequence for sheaves.

Proposition 4. The preshaf U 7! ker('(U)) is a sheaf and U 7! im('(U)) is not necessarily a sheaf.Moreover, ' is injective (surjective) if and only if the restriction

'x

: Fx

! Gx

is injective (surjective) for all x.

Remark 4. If ' : F ! G is surjective as a sheaf morphism, the morphism '(U) : F(U) ! G(U) is notnecessarily surjective.e.g. taking the quotient of the ideal sheaf of two points on P1.

Definition 4.2.2. A sequence of morphisms of sheaves

0 ! F '�! G �! H ! 0

is exact if ' is injective, is surjective and ker = im'.

Example 4.2.3 (Exponential exact sequence). Let X be a complex manifold. Let O⇤X

be the sheaf ofinvertible holomorphic functions. Then the exponential functions gives a sheaf homomorphism

exp : OX

! O⇤X

,

f 7! ef

Moreover, we can obtain an exact seuquence

0 ! Z 2⇡i��! OX

exp��! O⇤X

! 0,

called the exponential exact sequence.

Definition 4.2.4. A complex of sheaves is a collection of sheaves F i, together with morphisms of sheavesdi

: F i ! F i+1 such that di+1

� di

= 0. A resolution of a sheaf F is a complex F• and an injective mapj : F ! F0 such that the sequence

F0 ! F1 . . . ! Fk !is exact in the middle and ker(d

0

) = im(j).

4.2. RESOLUTIONS 31

Definition 4.2.5. A sheaf I over X is called injective if for every injective morphism i : G ,! H and everymorphism ' : G ! I, there exists an extension : H ! I such that � i = '. We say a resolution I• of Fis an injective resolution if every Ii is injective.

Proposition 5. For every sheaf F of abelian groups over X, there exists an injective sheaf I with an injectivemorphism F ,! I. In other words, every sheaf admits an injective resolution.

Proof. This follows from the following lemma:

Lemma 1. For every abelian group A, there exists an injective abelian group I with an injective homomor-phism A ,! I.

Admitting this fact, we can thus embedd F into the injective sheaf defined by

U 7!M

x2UIx

,

where Ix

is the associated injective abelian group in Lemma 1 for Fx

.

Sketch of the proof of Lemma 1. This is based on the following two simple facts.

(a) abelian group is injective if and only if it is divisible.

(b) quotient of injective abelian groups is injective.

Then we can embedd A into the quotient of the injective group Q(A), where Q(A) is the free Q-module onA. The assertion then follows from (a) and (b).

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Cech resolution

Let F be a sheaf over X and {Ui

} an open cover of X. We define

FI

:= (jI

)⇤(F|UI )

where UI

=Ti2I

Ui

and jI

: UI

,! X is the open immersion. We then define

Fk =M

|I|=k+1

FI

and d : Fk ! Fk+1 by the formula

(ds)j0,...,jk+1 =

X

i

(�1)isj0,...,ˆji,...,jk+1

|U\Uj0,...,jk+1

,

which is valid for s = (sI

), sI

2 FI

(U).

Proposition 6. The complex

0 ! F0

d�! F1

d�! F2 · · ·is a resolution of F .


Proof. It su�ces to check the exactness on the stalks. For x 2 Ui

, we define

� : Fk

x

! Fk�1

x

as below:(�s)

i0,...,ik�1 = sign(i, i0

, . . . , ik�1

)�i,i0,...,ik�1 ,

with �i,i0,...,ik�1 = 0 if i = i

j

for some 0 j k � 1. Here s is represented by (sI

2 F(VI

\ UI

)) for someopen neighbourhood V

I

of x. We can further assume that VI

✓ Ui

.

First, we shall show the map makes sense, which means that the right hand term is a germ of a section ofji0,...,ik�1⇤F . But this follows directly from the fact V

i,i0,...,ik�1 \ Ui0,...,ik�1 = V

i,i0,...,ik�1 \ Ui,i0,...,ik�1 .

Next, we haved � � + � � d = id.

This implies the exactness as desired.

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de Rham resolution

Let X be a smooth manifold. Let Ak

X

be the sheaf of smooth sections of the bundle ⌦k

X

. By Poincare lemma,the complex

0 ! A0

X

d�! A1

X

! · · · ! An

X

! 0

is a resolution of the constant sheaf R, called the de Rham resolution.

Dolbeault resolution

Let X be a complex manifold with a holomorphic vector bundle E ! X. Let E be the associated sheaf offree O

X

-module and let A0,q(E) be the sheaf of smooth sections of ⌦0,q ⌦ E. Then the complex

0 ! A0,0(E)¯

@�! A0,1(E) · · · A0,n(E) ! 0

is a resolution of the sheaf E by @-Poincare lemma.

4.3 Cohomology of Sheaves

Definition 4.3.1. Let F be a sheaf on M . Let F• be an injective resolution of F . We consider the functor�, which is defined by taking the global sections of sheaves. The i-th cohomology of F is defined to be thei-th cohomology of the complex �(F•).

Theorem 18. 1. H•(F) := H•(X,F) is independent of the choice of F•.

2. For every short exact sequence0 ! F ! G ! H ! 0,

there is a long exact sequence

0 ! H0(F) ! H0(G) ! H0(H) ! H1(F) ! · · ·

4.3. COHOMOLOGY OF SHEAVES 33

Proof. (1) We say two resolutions (I•, dI

), (J •, dJ

) are homotopy equivalent if there exists collections ofmorphism

f : I• ! J • and g : J • ! I•

such that g � f (resp. f � g) are homotopy equivalent to the identity, i.e. there exists a homotopy

H : I• ! I•�1 (resp. J • ! J •�1)

satisfyingH � d+ d �H = f � g � idI .

Obviously, if two resolutions are homotopy equivalent, then the cohomology of the complexes �(I•) and�(J •) are isomorphic. So it su�ces two show any two injective resolutions are homotopy equivalent.

Let us first construct a morphismf : I• ! J •

as below: the morphism f0 : I0 ! J 0 is obtained as the extension of F ! J 0 from the injectivity of J 0.

Now we can proceed by induction. For k � 1, once fk�1 is defined, the morphism fk : Ik ! J k is obtainedas the extension of the composition

coker(Ik�2 ! Ik�1) ! J k�1 ! J k.

Conversely, one can obtain a morphism g : J • ! J •. To show that f � g is homotopy equivalent to theidentity idJ , we define the homotopy

Hk : J k ! J k�1

defined as the extension of the map hk : coker(dk�2

J ) = im(dk�1

J ) ! J k�1 induced by

fk�1 � gk�1 � idk�1

J � dk�2

J �Hk�1 : J k�1 ! J k�1

The morphism hk is well defined because by induction, the kernel of the map above contains the image ofdk�2

J . Similarly, one can construct a homotopy from g � f to idI .

(2) This relies on the following lemma

Lemma 2. If

0 ! A f�! B g�! C ! 0

is a short exact sequence, then there exists injective resolutions I•,J •,K• such that we have an exact sequenceof complexes

0 ! I• ! J • ! K• ! 0.

Proof. Choose any injective resolution I• of A and K• of C. Note that A admits two resolutions

0 ! A ! B ! K0 ! · · ·0 ! A ! I•

which induces a morphism ' between two complexes (the proof is similar as in (1)). Then we can constructan injective resolution J • with J k = Ik �Kk defined as below:

j = (�'0, dK

� g0) : B ! J 0


anddnJ (x, y) = (dnI(x) + (�1)n'n+1(y), dK(x)).

This gives the result as desired. Now we have a short exact sequence (actually split) of complexes. Moreover,after taking the functor �(�). the complexes

0 ! �(I•) ! �(J •) ! �(K•) ! 0.

remains exact. Then the assertion follows from the snake lemma as below:

Lemma 3. For a commutative diagram of two exact sequences of abelian groups

A //

a

✏✏

B //

b

✏✏

C //

c

✏✏

0

0 // A0 // B0 // C 0

there is an exact sequence

ker a �! ker b �! ker cd�! coker a �! coker b �! coker c

Proof. Here we only explain how to define the map d by chasing diagram. Pick an element x 2 ker c. It canbe lifted to an element y 2 B. Because of the commutativity of the diagram, we know that the image of yin B0 lies in the kernel of B0 ! C 0 and hence the image of A0 ! B0. Thus we can find a preimage z 2 A0.Then we define d(x) = ⇡(z) where ⇡ : A0 ! coker a is the quotient map. |

Acyclic sheaves and acyclic resolution

We say that a sheaf F over X is acyclic if H i(X,F) = 0 for all i > 0.

Proposition 7. A sheaf F is said to be flasque if ⇢UV

is surjective for all V ✓ U . Then all falsque sheavesare acyclic.

Proof. We can embedd F into an injective sheaf I, which gives arise to a short exact sequence

0 ! F ◆�! I ⇡�! Q ! 0.

Then we have a long exact sequence

0 ! H0(X,F) ! H0(X, I) ! H0(X,Q) ! · · ·

We claim that the map H0(X, I) ! H0(X,Q) is surjective. This is because for every s 2 H0(X,Q), we canfind an open cover {U

i

} such that s|Ui can be lifted to an element t

i

in I(Ui

). The di↵erence

ti

|Uij � t

j

|Uij = ◆(r

ij

)

for some rij

2 F(Uij

). As F is flasque, we can take a lift rj

2 F(Uj

) whose restriction to Uij

is rij

. Let

vi

= ti

and vj

= tj

+ ◆(rj

),


which agrees on Uij

. So there exists v 2 I(Ui

[ Uj

) s.t. v|Uj = v

i

, v|Uj = v

j

, and ⇡(v) = s|Ui[Uj .

LetA = {V ✓ U open |9t 2 I(V ) with ⇡(V )(t) = s|

V

}.For each chain V

1

✓ V2

· · · of A, there is wi

s.t.

wi

2 I(Vi

) and ⇡(Vi

)(wi

) = s|Vi ,

and wj

|Vj�1 = w

j�1

. ThenS

Vi

2 A and is the upper bound of this chain. By Zorn’s lemma, there exists amaximal W and v0 2 I(W ) s.t. ⇡(W )(v0) = s|

W

. From the above proof, we know W = U .

Now, as the middle terms H i(X, I) = 0 for i > 0, we get H i(X,F) = H i+1(X,Q) for all i > 0. Note that Qis also flasque, we can prove the assertion by induction on i.

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Why acyclic sheaves are important?

Theorem 19. If F has a acyclic resolution M•, then H i(F) = H i(�(M•)).

Proof. The proof is by induction on i (for any sheaf). Note that we have a short exact seuquence

0 ! F ! M0 ! Q ! 0

which induces a long exact sequence

! H i(F) ! H i(M0) ! H i(Q) ! H i+1(F)

The quotient sheaf Q has a shifted resolution M•�1. Clearly, we have

H1(F) = Coker(�(M0) ! �(Q)) = ker(�(M1) ! �(M2))/im(�(M0) ! �(M1))

by definition. Moreover, when i � 1, as M0 is acyclic, we have H i+1(F) = H i(Q). Then by induction weget

H i+1(F) = H i(Q) = H i(�(M•�1)) = H i+1(�(M•)).

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Example 4.3.2. The constant sheaves on varieties with Zariski topology are flasque and hence acyclic.

Proposition 8. Let A be a sheaf of rings over X satisfying that for every open cover {Ui

}, there exists apartition of unity f

i

2 �(A) withP

fi

= 1, subordinate to this open cover. Then we say sheaves of A-modulesare fine.

Proof. First, F admits a flasque resolution (I•, d) and we have Hk(F) = Hk(�(I•, d)). Let ↵ 2 ker(d). Thelocal exactness of the complex gives there exists an open cover {U

i

} such that

↵|Ui = d(�

i

)

for some � 2 Ik�1(Ui

). Set

� =X

fi

�i


where fi

are the partition of unity subordinate to {Ui

}. Note that fi

�i

can be viewed as a section on Ik�1

and it is zero outside Ui

. Then we know

↵ =X

fi

↵|Ui = d(�)

is exact. This proves the assertion. |

Since Ak

X

are sheaves of C1X

-modules and therefore fine, we get

Corollary. Let X be a smooth manifold. Then H i(X,R) = H i

dR

(X,R).

de-Rham Theorem

Theorem 20 (de-Rham). There is an isomorphism H i

dR

(M,R) ⇠= H i

sing

(M,R).

Proof. Let us consider the sheaf Cq

sing

of singular chains over X defined as the sheafification of the presheaf

Cq

sing

(U) = {singular q-cochains on U}

We can get a complex of sheaves

C0

sing

@�! C1

sing

@�! C0

sing

· · ·with the usual boundary map @. Moreover, it is a resolution of the constant sheaf Z because locally, the

singular cohomology of contractible open subsets are zero, and ker(C0

sing

(X)@�! C1

sing

(X)) = Z.

Moreover, one can easily check the sheaf Cq

sing

is flasque and hence acyclic. So we have H i(X,Z) =

H i(�(C•sing

)). It remains to show that two complexes �(C•sing

) and C•sing

(X) are quasi-isomorphic. Thisfollows from a theorem of Spanier.

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Cech cohomology

Let U = {Ui

} be a open cover of X. Let H(U,F) be the q-th cohomology group of the Cech complex

Cq(U,F) =M

|I|=q+1

F(UI

).

This is called the Cech cohomology of F w.r.t. U.

Theorem 21. If Hq(UI

,F) = 0 for all q > 0, then

Hq(F) = Hq(U,F).

Sketch of the Proof.

e.g. X = C2 � {0} as an algebraic variety. Take an open cover U1

= {x 6= 0} = Spec C[x, y]x

andU2

= {y 6= 0} = Spec C[x, y]y

. The corresponding Cech resolution of OX

is

! C0(U,OX

)@�! C1(U,O

X

) ! 0


where C0(U,OX

) = C[x, y]x

� C[x, y]y

, and C1(U,OX

) = C[x, y]x,y

. So

H1(OX

) = ker(@) =⌦xiyj | i, j < 0

↵

is infinite dimensional.

Invertible sheaves, line bundles

Theorem 22. Let A⇤ be the sheaf of invertible elements in a sheaf A of rings. Then

H1(X,A⇤) = {sheaves of rank one free A-modules}/ ⇠== {line bundles with flat/holomorphic structure w.r.t. A}/ ⇠= .

Proof. This can be explained using Cech cohomology. Roughly speaking, the transition functions of thetrivialization are lying in C1(U,A⇤) satifying the cocycle condition, which means they actually lie in thekernel of the Cech di↵erential. This exactly represents an element in H1(U,A⇤) = H1(X,A⇤). |

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