maperez.netmaperez.net/documents/Complex_Geometry.pdf · TABLE OF CONTENTS 1 COMPLEX ANALYSIS 1 1.1 Complex Analysis in one variable . . . . . . . . . . . . . . . . . . . . .

MARCO A. PEREZ B.Universite du Quebec a Montreal.Departement de Mathematiques.

COMPLEX GEOMETRY

Course notes

December, 2011.

These notes are based on a course given by Steven Lu in Fall 2011 at UQAM. All errorsare responsibility of the author.

On the cover: a picture of the Riemann Sphere

(taken from: http://en.wikipedia.org/wiki/Riemann sphere).

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ii

TABLE OF CONTENTS

1 COMPLEX ANALYSIS 1

1.1 Complex Analysis in one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Complex Analysis in several variables . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 RIEMANN SURFACES 9

2.1 Complex manifolds, Lie groups and Riemann surfaces . . . . . . . . . . . . . . . . 9

2.2 Holomorphic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Meromorphic functions and differentials . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Weierstrass P -function on C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Dimension on Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Covering spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7 The Riemann surface of an algebraic function . . . . . . . . . . . . . . . . . . . . . 20

2.8 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.9 Topology of Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.10 Product structures on⊕

iHidR(Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.11 Questions about (compact) Riemann surfaces . . . . . . . . . . . . . . . . . . . . . 31

2.12 Harmonic differentials and Hodge decompositions . . . . . . . . . . . . . . . . . . . 32

2.13 Analysis on the Hilberts space of differentials . . . . . . . . . . . . . . . . . . . . . 34

2.14 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.15 Proof of Weyl’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.16 Riemann Extension Theorem and Dirichlet Principle . . . . . . . . . . . . . . . . 39

2.17 Projective model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.18 Arithmetic nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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3 COMPLEX MANIFOLDS 43

3.1 Complex manifolds and forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Kahler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Metrics and connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5 The Fubini Study metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 SHEAF COHOMOLOGY 53

4.1 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Cohomology of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3 Coherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 HARMONIC FORMS 63

5.1 Harmonic forms on compact manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Some applications of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 Heat equation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.5 Index Theorem (Heat Equation approach) . . . . . . . . . . . . . . . . . . . . . . . 71

BIBLIOGRAPHY 73

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Chapter 1

COMPLEX ANALYSIS

1.1 Complex Analysis in one variable

Let U ⊆ C = R2 be an open subset of the complex plane. We shall denote an element z ∈ C by z = x+ iy,where i =

√−1. An function f : U −→ C is holomorphic on U if it is complex differentiable at all points

of U , i.e.,

f ′(z0) = dfdz (z0) = limz→z0

f(z)−f(z0)z−z0

exists for every z0 ∈ U . We shall denote this by f ∈ O(U). If S ⊆ C is any subset, we shall say that f isholomorphic on S (f ∈ O(S)) if f is holomorphic on a open neighbourhood of S.

If the function f is R-differentiable on U then ∂f∂xdx + ∂f

∂y makes sense and df(x, y) ∈ HomR(Tz=x+iyU,R2).Recall that

dz = dx+ idy and dz = dx− idy

Using these expressions, we can write the differential df as

df = 12

(∂f∂x − i

∂f∂y

)dz + 1

2

(∂f∂x + i∂f∂y

)dz = ∂f

∂z dz + ∂f∂z dz

Notice the following relations(∂f∂z

)= ∂f

∂z and(∂f∂z

)= ∂f

∂z

Recall that

f is complex differentiable⇐⇒

f is R-differentiable and ∂f∂z = 0 (Cauchy-Riemann condition)⇐⇒

∂u∂z = −i∂v∂z⇐⇒

df =

(ux uyvx vy

)is a rotation matrix up to a real scalar multiple (ux = vy and uy = −vx).

1

We shall denote V ⊂⊂ U if V ⊆ U is compact (V is precompact in U) and ∂V is rectifiable, i.e., ∂V ispiecewise smooth.

Theorem 1.1.1 (Cauchy). f ∈ O(U) if and only if∫∂V

f = 0, for every V ⊂⊂ U simple connected.

Theorem 1.1.2 (Cauchy’s Integral Formula). z0 ∈ V ⊂⊂ U if and only if

f(z0) =1

2πi

∫

∂V

f(z)

z − z0dz.

If V = Dε(z0) is a disk centered at z0 of radius ε, then we shall denote the previous integral by

f(z0) = Avg∂V (f) :=1

2π

∫ 2π

0

f(z0 + εeiθ)dθ.

Corollary 1.1.1 (Liouville Theorem). Every holomorphic function on C is constant.

Proof: Let Vε = Bε(z0), z0 ∈ C. We show that f ′(z0) = 0. Using the Cauchy’s Integral formula, wehave

f ′(z0) =1

2πi

∫

∂Vε

f(z)

(z − z0)2

Notice that |f | is bounded on ∂Vε. Then |f(z)| ≤M on ∂Vε for some M > 0. So we have

|f ′(z0)| = 1

2π

∣∣∣∣∫

∂Vε

f(z)

(z − z0)2dz

∣∣∣∣ ≤1

2πε2

∫

∂Vε

|f(z)||z − z0|2

dz

=1

2πε2

∫

∂Vε

|f(z)|dz ≤ M

2πε2

∫

∂Vε

dz

=M

2πε2· 2πε =

M

ε

It follows |f ′(z0)| −→ 0 as ε −→∞. Hence f ′(z0) = 0 for every z0 ∈ C and f is constant in C.

Corollary 1.1.2 (Riemann Extension Theorem). If f ∈ O(U − z0), bounded near z0 and continuous at z0,then f ∈ O(U).

Proof: If f ∈ O(U − z0) then f ∈ O(U − Bε(z0)), for some ε > 0. Then the result follows since theCauchy’s Integral Formula still holds in this case.

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Theorem 1.1.3 (Local Structure of f ∈ O(U)). If f ∈ O(U) is non-constant at z0 ∈ U . Let

m = minn > 0 / f (n)(z0) 6= 0.

Then there exists a bi-holomorphic function ϕ : V −→W from a neighbourhood of V of z0 to a neighbourhoodW of 0 ∈ C with ϕ(z0) = 0 such that

f(z)− f(z0) = ϕ(z)m, for every z ∈ V.

Proof: Note that f(z) − f(z0) = (z − z0)mg(z) with g(z0) 6= 0 and g ∈ O(U). Since the quotientf(z)−f(z0)

z−z0 is bounded on U − z0 and continuous at z0, we have by the Riemann Extension Theorem that

(z − z0)m−1g(z) = f(z)−f(z0)z−z0 is holomorphic on U . Proceeding this way, we have that g(z) ∈ O(U).

We study several cases: If n = 1 then f ′(z0) 6= 0 and by the Inverse Function Theorem we can chooseϕ(z) = f(z) − f(z0). Now assume n 6= 1. Since g(z0) 6= 0 then g(z) 6= 0 on a neighbourhood of z0. Sowe can write g = hm on a neighbourhood V . We have

f(z)− f(z0) = [h(z)(z − z0)]m

with ϕ′(z0) = h(z0) 6= 0. Hence, up to a local change of coordinates, f is locally of the form z 7→ zm forsome m. Such a number m is called the ramification degree of f at z0.

Corollary 1.1.3 (Open Mapping Theorem). If f ∈ O(U) is non-constant and U is connected, then f is anopen mapping.

Corollary 1.1.4. If f ∈ O(U) and |f | has a local maximum at z0 ∈ U , where U is an open connected set,then f is constant on U .

Proof: Suppose f is not constant. Then by the Open Mapping Theorem, we have that Bε(f(z0)) ⊆ f(U)for some ε > 0. In this neighbourhood there are some points of modulus greater that 0. Hence f(z0) isnot a local maximum.

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1.2 Analyticity

A function f : U −→ C is said to be real analytic on U if for every z0 = (x0, y0) ∈ U there exists aneighbourhood V of z0 such that

f(z) =∑∞α,β=0 aα,β(x− x0)α(y − y0)β for every z ∈ V.

Similarly, f is said to be complex analytic on U if for every z0 ∈ U there exists a neighbourhood V of z0

such thatf(z) =

∑∞n=0 an(z − z0)n for every z ∈ V.

In both cases the equality means normal convergence in U , i.e., uniform convergence on compacts in U .

Theorem 1.2.1. f ∈ O(U) if and only if f is complex analytic on U .

Proof: We know that

f(z) =1

2πi

∫

∂Dr(z0)

f(w)

w − z dw.

On the other hand,

1

w − z =1

w − z

(1

1− z−z0w−w0

)=

1

w − z0

∞∑

n=0

(z − z0

w − z0

)n.

It follows that

f(z) =1

2πi

∫

∂Dr(z0)

f(z)

∞∑

n=0

(z − z0)n

(w − z0)n+1dw =

∞∑

n=0

an(z − z0)n

where

an =1

2πi

∫

∂Dr(z0)

f(w)

(w − z0)n+1dw =

1

n!f (n)(z0)

and |w − z0| = r on ∂Dr(z0).

Theorem 1.2.2. If f ∈ O(U) is non-constant, where U is connected, then f−1(0) is discrete in U .

Proof: Suppose f−1(0) is not discrete. Let γ be an isolated point in f−1(0) and consider the Taylorexpansion of f about γ,

f(z) =

∞∑

n=0

f (n)

n!(z − γ)n

for every z ∈ Dr(γ), where r is the radius of convergence of the series. Since γ is not isolated in f−1(0),there exists z ∈ f−1(0) ∩Dr(γ). We have

0 =

∞∑

n=0

f (n)

n!(z − γ)n

4

and so f (n)(γ) = 0 for every n ≥ 0. It follows that f is constant on an neighbourhood of γ, getting acontradiction.

Corollary 1.2.1 (Analytic Continuation or Identity Theorem). If f = g on a non-discrete subset of U andf, g ∈ O(U), then f ≡ g on U .

Lemma 1.2.1 (Schwartz). If f ∈ O(D) and f ≤M on ∂D, |f(z)| ≤M |z| on ∂Dr for every r ∈ (−ε, 1), thenf(z) = Nz; where |N | < M .

Another version: If |f(z)| ≤ M on D and f(0) = 0, then f(z) = Nz with |N | < M . Here, Ddenotes the Poncare disk, i.e., the disk z ∈ C : |z|2 < 1 endowed with the metric

δ(z, w) = 2|z − w|2

(1− |z|2)(1− |w|2).

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1.3 Complex Analysis in several variables

Let U ⊆ Cn be an open subset.

Definition 1.3.1. A function f : U −→ C in C ′R(U) (R-differentiable on U) is called holomorphic, denotedf ∈ O(U), if for every u ∈ U , the differential dfu ∈ Hom(TuU,C) is a C-linear map.

We prove as before the following result:

Theorem 1.3.1. The following conditions are equivalent:

(1) f ∈ O(U).

(2) For every z0 ∈ U , f has the form

f(z0 + δ) =∑

I

aIδI (normal convergence)

where I = (i1, . . . , in) and zI = zi11 · · · zinn .

(3) If D = (ζ1, . . . , ζn) / |ζi−ai| < αi ∈ R>0 is a poly-disk in U , then for every z = (z1, . . . , zn) ∈ Int(D)

(1

2πi

)n ∫

|ζi−ai|=αif(ζ)

dζ1ζ1 − z1

∧ · · · ∧ dζnζn − zn

where δD = |ζi − ai| = αi ⊂ ∂D.

Note that if f ′ ∈ C ′R(U) then

df =∑ ∂f

∂xidxi +

∑ ∂f∂yi

dyi = 12

∑(∂f∂xi− i ∂f∂yi

)dzi + 1

2

∑(∂f∂xi

+ i ∂f∂yi

)dzi

Theorem 1.3.2. Let f ∈ Cn+1 such that f(0) = 0. Write Cn+1 = (w, z1, . . . , zn) = (w, z). If f 6≡ 0 onthe w − axis (z = 0) then on some neighbourhood V of 0, we have

f = (w, z)(wd + a1(z)wd−1 + · · ·+ ad(z))

where g is never zero on V .

Denote p(w, z) = wd + a1(z)wd−1 + · · · + ad(z). Hence locally we have zero(f) = zero(p). It follows thatthe roots of p are single valued holomorphic functions w = bi(z) away from the discriminant locus of f(∆f (z) = 0, where ∆f (z) is a polynomial in the ai’s). Hence f = 0 is a etale cover that covers the hyper-plane w = 0 = (z1, . . . , zn). So by induction we see that:

Fact 1.3.1. The zero of a holomorphic function is the disjoint union of submanifolds of lower dimension.

Definition 1.3.2. An analytic set is the set of common zeros of finitely many analytic functions.

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Theorem 1.3.3 (Riemann Extension Theorem).

• Part I: If f is a holomorphic function and bounded outside an analytic subset of codimension 2 orhigher, then f extends over the subset as a holomorphic function.

• Part II (also known as the Hartogs Theorem):

(1) Let U = ∆(r) = (z1, z2) / |z1| < r and |z2| < r and V = ∆(r′), such that r′ < r and V ⊂⊂ U ,then every f ∈ O(U − V ) extends to a holomorphic function on U .

(2) If S ⊆ U ⊆ Cn has complex codimension greater or equal than 2, where S is an analytic subsetand f ∈ O(U − S), then f extends to a holomorphic function on U .

Proof: We only proof the second part.

(1) Take a slice z1 = const. Then U − V = r′ < |z2| < r on this slice. Set

F (z1, z2) =1

2πi

∫

|w2|=r

f(z1, w2)

w2 − z2dw2

Hence F : U −→ C is holomorphic in z1 since

∂f

∂z1= 0 =⇒ ∂F

∂z1

and clearly also in z2 (Cauchy’s Integral Formula). Moreover, F = f on U − V by the Cauchy’sIntegral Formula.

(2) The a 2-dimensional slice and apply (1).

Theorem 1.3.4 (Open Mapping Theorem). If f ∈ O(U) then f is open.

Theorem 1.3.5 (Maximum Principle). |f | has no local maximum unless it is locally constant there.

Theorem 1.3.6 (Analytic continuation). f = 0 in an open subset of U and f ∈ O(U), where U is connected(or arcwise connected), then f ≡ 0 on U .

Proof: For every path α, the set I = c / f α(t) = 0 ∀t < c is open and closed.

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Chapter 2

RIEMANN SURFACES

2.1 Complex manifolds, Lie groups and Riemann surfaces

Definition 2.1.1. A complex manifold M is a topological manifold whose coordinate charts are opensubsets of Cn, such that the transition maps are holomorphic. The number n is called the dimension of M .A Riemann surface is a complex manifold of dimension 1. A complex Lie group is a group that is acomplex manifold such that the product and inversion maps are holomorphic.

Remark 2.1.1. Manifolds are always connected unless otherwise specified.

Example 2.1.1. The following sets are complex manifolds:

(1) CPn = [z0 : · · · : zn] = Cn+1 − 0/ ∼, where

(z0, . . . , zn) ∼ (z′0, . . . , z′n)⇐⇒ z = tz′, for some t ∈ C∗.

Let U0 = [z0 : · · · : zn] / z0 6= 0. Let ϕ0 : U0 −→ Cn be the map given by

[z0 : · · · : zn] 7→(z1

z0, . . . ,

znz0

)

which we shall call the 0-th affine chart. Note that there exist n+ 1 affine charts that cover CPn.

(2) The compact complex torus Cn/Γ, where Γ ∼= Zn.

(3)

PGL(1, 0) = Aut(CP1) = set of Moebius transformations

=

az0 + bz1

cz0 + dz1/ ad− bc 6= 0

/±1

The following sets are Riemann surfaces:

(4) CP1.

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(5) C and C∗ = C− 0.

(6) The half-plane model H = z / Im(z) > 0.

(7) The Poincare disk model ∆ = D = z / |z| < 1 and ∆∗ = ∆− 0. The models H and D are relatedby the map

z 7→ z − az − a

The exponential function exp : C −→ C∗ is a universal covering.

Definition 2.1.2. The manifold C/Γ (and more generally its nontrivial holomorphic images) is called anelliptic curve. The manifold CP1 is called a rational curve.

Note that CP1 has positive curvature, C has zero curvature (in other words, C is said to be flat), and H andD have negative curvature. Note that Γ is a lattice n + α / α ∈ H. The parameter space of ellipticcurves is the quotient H/SL(2,Z). Note that C/Γ has genus 1.

Example 2.1.2.

(1) Let f(z0, . . . , zn) be a homogeneous polynomial. Then

C = V (f) = [z0 : z1 : z2] / f(z0, z1, z2) = 0 ⊆ CP

is called an algebraic plane curve over C. This cuve C is smooth (or non-singular) if it is asubmanifold (only need to check df(p) 6= 0 for every p ∈ C to have C smooth).

(2) xd + yd + zd gives the Fermat curve of degree d in CP2. It is smooth since df 6= (0, 0, 0) on C,df = (xd−1dx, yn−1dy, zn−1dz).

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2.2 Holomorphic maps

Definition 2.2.1. Let f : X −→ Y be a continuous map of topological spaces, the inverse image f−1(p) iscalled the fibre of f at p ∈ Y . The map f is called discrete if all its fibres are discrete in X.

Theorem 2.2.1 (Identity Theorem). If f1, f2 : S1 −→ S2 are mappings of Riemann surfaces such that theycoincide on a non-discrete subset of S1, then f1 ≡ f2.

Theorem 2.2.2. Any non-constant mapping of Riemann surfaces is discrete.

The Open Mapping Theorem implies the following result:

Theorem 2.2.3. Let f : S1 −→ S2 be a non-constant map of Riemann surfaces. Assume that S1 iscompact. Then f is surjective and S2 is compact. Furthermore, if f is proper (i.e., f−1(C) is compactfor every compact set C) and discrete with finite fibres (i.e., a finite map) then f is called a branchedcovering map (at a branch f looks like z 7→ zd, where d is called the branched degree).

Let p ∈ S be a ramification point. At such a point we call the multiplicity (or the ramification degree)of f

multp(f) = d.

The degree of f is defined by

deg(f) =∑p∈F multp(f)

for every fibre F . The ramification index of f at p is

rp(f) = multp(f)− 1.

A map is said to be unramified if rp(f) = 0 for every p ∈ S.

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2.3 Meromorphic functions and differentials

Definition 2.3.1. A meromorphic function on a Riemann surface Z is a holomorphic function on anopen subset U ⊆ Z where Z − U is a discrete set consisting of at most poles of the function.

Recall that a pole p ∈ Z − U is defined by one of the following equivalent conditions:

(a) limz→pf(z) =∞.

(b) f can be written locally as a Laurent series

f(z) =

∞∑

−∞aiz

i

with ai = 0 for every i < n ∈ Z.

(c) f = g/h, where g, h ∈ O(p), g(p) 6= 0 and h(p) = 0.

The set of such functions is denoted M(Z). We have

f ∈M(Z)⇐⇒ f : Zhol−→ CP1

and thatpoles of f = f−1(∞)

Example 2.3.1.

(1) A non-constant polynomial defines a meromorphic function from CP1 with pole order at ∞ equal todeg(f) ≥ 1.

(2) A rational function p(z)/g(z) defines a meromorphic function with pole order at ∞ equal to deg(p)−deg(g). If this difference is negative then f has a zero at ∞.

Fact 2.3.1. M(Z) is a field.

A finite map of Riemann surfaces f : Z1 −→ Z2 corresponds to a finite field extension

f∗ :M(Z2) →M(Z1).

Definition 2.3.2. A meromorphic differential on a Riemann surface Z is a holomorphic differential ωon an open U ⊆ Z whose complement Z − U is discrete and consist of poles of ω. Locally, ω = fdz even ata pole. The pole order of ω is defined by that of f (locally) and its residue at p is the same as that of fdz(p = 0), denoted Resp(ω).

Theorem 2.3.1 (Residue). V ⊂⊂ Z with rectifiable boundary ∂V and ω differentiable on Z. Then

∫

∂V

ω =∑

p∈VResp(ω)

12

We shall denote the space of meromorphic differentials by M′(Z).

Theorem 2.3.2. ω ∈M′(Z),∑p∈Z Resp(w) = 0.

Corollary 2.3.1 (Sum rule or product rule, Reciprocity Theorem). If Z is compact and f ∈M(Z), then

#zero(f) = #poles(f)

on Z counting multiplicity.

Corollary 2.3.2. If f ∈M(CP1) then f is rational. Hence M(CP1) = C(Z).

Corollary 2.3.3. M′(CP′) = C(Z)dz.

Let S be a Riemann surface and U ⊂⊂ S a relative compact open subset of S with good ∂U . Let ω be ameromorphic differential (ω ∈M′(S)). Then we have

∫

∂U

ω =∑

p∈URespω

If ∂U = ∅ (so U = S is compact) then ∑

p∈SRespω = 0

We have that if S is compact then S = R(S), the set of rational functions (later, we are going to study thisin detail).

Example 2.3.2. M(P1) = R(P1), where P1 = CP1. Note that CP1 is compact and CP1 = C ∪ C, wherethere is a map between charts

z ∈ Cw= 1

z7→ w ∈ C

Since∑p∈S Respω = 0, we have ω = df

f , and Respω = ordpf .

Recall that f ∈M(P1) if and only if f : P1 −→ P1 is holomorphic.

Example 2.3.3. Let C be a quadratic (conic) curve in CP2 = P2. Let p ∈ C. The space of lines passingthrough p is P2 and so gives a meromorphic map P2 −→ P1. We have the following diagram:

P2 P1

P2 − p

C − p

hol

hol

13

This shows that C ∼=hol P1. For example, we have the Fermat curve z20 + z2

1 = z22 . If P1 = [u : v], then

define a map z0 = (u− v)2, z1 = 2uv and z2 = (u+ v)2.

Note that π1(S) = 0 where S = P1,C,D. Recall that if π1(S) = Z then S is not compact.

Example 2.3.4. S1 = C∗, S2 = D − 12D. These two examples are not biholomorphic Riemann surfaces.

Neither D∗ is biholomorphic to C∗. If so, then a biholomorphic function D∗ −→ C∗ produces an extensionD −→ S, getting a contradiction.

The Riemann surfaces H and D are biholomorphic via the map

ω 7→ w − aw − a

14

2.4 Weierstrass P -function on C

Consider the torus C/Γ, whereΓ = 〈1, τ〉τ∈C = Z⊕ Zτ

is a lattice in C. Define the Weirstrass P -function by the formula:

P(z) = p(z,Γ) = 1z2 +

∑λ∈Γ−0

[1

(z−λ)2 − 1λ2

]

Hence P ∈M(C) and satisfies:

(a) P(z) = P(−z),(b) P(z + λ) = P(z) for every λ ∈ Γ,

(c) there exists no other poles.

Hence P descends to a meromorphic function on C = C/Γ with a pole at 0 (degree = 2), i.e., we have aholomorphic function

Cf−→ CP1

Locally, this map looks like z 7→ zn about 0. In this case n = 2. This map has exactly degree 2 since thereare no other poles. Moreover, f is branched at 0.

∞ CP1

0

Now P ′(z) is also periodic (period Γ) with triple poles on Γ and no other poles. The map C −→ P2 given by

z ∈ C 7→ [P(z) : P ′(z) : 1]

defines a holomorphic function C − 0 −→ P2,

[P(z) : P ′(z) : 1] =

[ P(z)

P ′(z) : 1 :1

P ′(z)

]

and it extends over 0.

15

2.5 Dimension on Riemann surfaces

Definition 2.5.1. A divisor D on a Riemann surface S is a formal Z-linear combination of points in S

D =∑aiPi

where ai 6= 0 for every i and Pi is a discrete subset of S. A divisor D is called effective if ai ≥ 0 for every i.

If supp(D) = Pi / ai 6= 0 is finite, then

deg(D) :=∑ai

Definition 2.5.2. Let f ∈M(S)− 0 =M∗(S). The divisor of f is defined by

(f) := (f)0 − (f)∞

where(f)0 =

∑(ordP f)P and (f)∞ =

∑P∈f−1(∞)(multP f)P

Note that multP f = −ordP f , so we can rewrite the previos expresion as

(f) =∑

(ordP f)P

Lemma 2.5.1. If S is a compact Riemann surface, then deg(f) = 0 for every f ∈ M∗(S), where deg is amap Div(S) −→ Z.

Definition 2.5.3. A divisor is called principal if it lies in the image of deg( ) :M∗ −→ Z.

Definition 2.5.4. Two divisors D1 and D2 are said to be linearly equivalent, denoted D1 ∼ D2, if D1−D2

is principal.

Example 2.5.1. D1 ∼ D2 on P1 if and only if deg(D1) = deg(D2).

Example 2.5.2. What condition we need if we want D1 ∼ D2 on C = C/Γ. Let p, q ∈ C be two distinctpoints in C and suppose that D = p− q = (f) for some f ∈M∗. Then f is a map C −→ P1 with deg(f) = 1.We have (f)0 = p and f is bijective. Then f is a biholomorphic map, getting a contradiction.

Given ω ∈M′(S)∗. Recall that this means ω = fdz for a local coordinate z at p and f ∈M(p).

Definition 2.5.5. ordpω = ordpf . The divisor of the form

(ω) :=∑p∈S(ordpω)P

is called a canonical divisor and is denoted KS or simply K. A divisor is canonical if D ∼ (w).

16

Example 2.5.3.

• KC/Γ = (dz) = 0 · P .

• KP1 = (dz) = −2 · ∞, z = 1ω , dz = −dωω2 .

Theorem 2.5.1 (Riemann - Hurewicz). If f : S1 −→ S2 is a finite map, then KS1 ∼ K + S2 + R where Ris the ramification divisor

R =∑

p∈S1

rp(f)P,

where each rp(f) ≥ 0.

Proof: Locally, f looks like z 7→ zn, and dzn = nzn−1dz.

Corollary 2.5.1 (Riemann - Hurewicz formula).

2 · g(S1)− 2 = degKS1= (degf) · degKS2

+ degR,

where KS1 = (ω), ω ∈ Γ(T∨S1) (global sections of the tangent bundle) ←→ ω∨ ∈ Γ(TS1).

17

2.6 Covering spaces

Recall that an etale space (or etale map) over X is a continuous map p : X −→ X such that p is a local

homeomorphism: that is, for every x ∈ X, there is an open set U in X containing x such that the imagep(U) is open in X and the restriction of p to U is a homeomorphism p|U : U −→ p(U). A connected covering

space p : X −→ X is a universal cover if X is simply connected. The name universal cover comes fromthe following important property: if the map p : X −→ X is a universal cover of the space X and the mapp′ : X ′ −→ X is any cover of the space X where the covering space X ′ is connected, then there exists acovering map f : X ′ −→ X such that p f = p′. Any manifold X has a universal cover X with etale coveringmap f : X −→ X and π1(X) acts on X discretely and freely with quotient f . Moreover, there exists abijection

π1(S) ⊇ H 7→ X/H etale−→ Xbetween the set of subgroups of π1(X) up to conjugation and the set of etale coverings from a connectedmanifold.

G/H for H normal ⇐⇒ Galois (regular) covering.

Recall that a covering map p : X −→ X is said to be Galois if for every x ∈ X and x ∈ p−1(x), the subgroup

p∗π1(X, x) is normal in π1(X,x).

If X is a Riemann surface, then the complex charts on X lifts to any covering space.

Lemma 2.6.1. Let f : Z1 −→ Z be a finite etale covering corresponding to H → π1(Z). Then there existsa finite regular covering h : Z2 −→ Z and a finite etale covering g : Z2 −→ Z1 such that f g = h.

Z1 Z2

Z

etale covering

∃regu

lar

Proof: H has a finite number of conjugates in π1(Z), such a number equals [π1(Z) : N(H)] and theseintersection is then of finite index.

Corollary 2.6.1. For every n, there exists a unique etale n-sheeted covering Z −→ D∗ and it is isomorphicto D∗ −→ D∗ (z 7→ zn).

Example 2.6.1. If f : Z1 −→ Z2 is finite between Riemann surfaces, then there is a finite etale covering

Z1 − f−1(∆) −→ Z2 −∆

where ∆ = f(suppRf ) is the branching locus.

18

Conversely, the previous corollary implies:

Theorem 2.6.1. Let ∆ ⊆ Z2 be a discrete subset. A finite etale covering U −→ Z2−∆, where U is an opensubset, has a unique continuation to a finite map

U ⊆ Z1 −→ Z2

where Z1 is a Riemann surface.

19

2.7 The Riemann surface of an algebraic function

Let Z2 be a Riemann surface.

Proposition 2.7.1. LetP (T ) = Tn + c1T

n−1 + · · ·+ cn

in M(Z2)[T ] be an irreducible polynomial. Then there exists a map of Riemann surfaces f : Z1 −→ Z2 ofdegree n and a meromorphic function F ∈M(Z1) that satisfies:

Fn + f∗(c1)Fn−1 + · · ·+ f∗(cn) = 0. (∗)

Proof: Let ∆ ⊆ Z2 be the discrete set containing the poles of the ci’s and the points p where

Pp(T ) := Tn + c1(p)Tn−1 + · · ·+ cn(p)

has multiple roots. Then U = (p, z) ∈ (Z2 −∆)× C / Pp(z) = 0 is a Riemann surface, and Z2 −∆ isa finite etale cover. We claim that it is connected, i.e.,

Claim: Given f : Z1 −→ Z2, then every F ∈ M(Z1) is algebraic over M(Z2) and satisfies an equalityof the form (∗) but with degree less or equal than the degf .

Corollary 2.7.1. If Z1 −→ Z2 is finite, then

f∗ :M(Z2) −→M(Z1)

is a finite field extension.

Example 2.7.1. M(P1) = C(Z), the field of rational functions in variable z.

Theorem 2.7.1. As soon as there exists a meromorphic function f on Z1 (←− compact =⇒ f is finite),M(Z1) is finite algebraic over C(z) of extension deg = degf .

Z1f−→ P1

Corollary 2.7.2.

(1) If Z1f−→ Z2 is finite then M(Z2)

f∗

→M(Z1) is a finite field extension of degree (f).

(2) Conversely, letM(Z2)ϕ→M(Z1) = K be a finite field extension of degree d. Then there exists a finite

map Z1 −→ Z2 of degree d whose field extension is isomorphic to ϕ.

(3) A field K of transcendental degree = 1 over C is isomorphic to M(Z) of some compact Riemannsurface. Such a Z is called a model of K.

20

2.8 Review

Note that the ratio of two meromorphic 1-forms is a meromorphic function:

ω1, ω2 ∈M′(Z) =⇒(ω1

ω2

)= (f) ∈ DivP (Z), where f ∈M(Z).

Here, DivP denotes the set of principal divisors. Note that ω ∈ M′(Z) if and only if ω ∈ Γm(T∨Z), whereT∨Z denotes the cotangent bundle of Z and Γm(T∨Z) is the set of holomorphic sections of T∨Z. Also,T ⊗ T∨ = O (trivial line bundle) and so

1 =

(1

ω

)· ω ∈ Γm(O) = O,

where 1ω ∈ TZ and ω ∈ T∨Z.

If ω1 = fω2 then (ω1) = (f) + (ω2). So we get the formula

deg( ) = deg(f) + deg( )

On the other hand, deg(f) = 0. To show this, we know that f(z) = zn locally, where n = ord(f). Thendff = ndzz and

∑p∈Z Resp

dff = 0, where Z is compact. Hence we get the following result:

Theorem 2.8.1. deg(ω) has the same value for any ω ∈ M′(Z), assuming that Z is a compact andconnected Riemann surface.

Let g be the topological genus of Z. We have the following relations:

deg(ω) = 2g − 2 and deg(

1ω

)= 2− 2g = χ(Z)

where 1ω ∈ Γ(TZ).

genus = # of holes

Recall the following results:

21

Theorem 2.8.2. Let P (T ) = Tn + c1Tn−1 + · · ·+ cn ∈M(Z)[T ] be an irreducible polynomial in T over Z.

Then there exists a finite map of Riemann surfaces f : Z ′ −→ Z of degree n, unique up to isomorphisms,and a meromorphic function F on Z ′ satisfying

Fn + (f∗c1)Fn−1 + · · ·+ (f∗cn) = 0

Corollary 2.8.1.

(1) Z1 −→ Z2 finite =⇒ f∗ :M(Z2)ϕ→M(Z1) is a finite field extension of degree n.

(2) Conversely, any finite field extension of degree n gives rise to a finite map Z1 −→ Z2 whose associatedfield extension is isomorphic to ϕ.

(3) A field extension K of transcendence degree 1 (i.e., K is a finite field extension over C(Z)) is isomorphicto M(Z) for some Riemann surface Z (finite over CP1).

Z is called a smooth model of K.

Theorem 2.8.3. Let C ⊆ CP2 be an irreducible plane curve, i.e., C = V (F ) where F is an irreduciblehomogeneous polynomial in (z0 : z1 : z2). Then there exists a compact Riemann surface Z and a genericallyinjective map f : Z −→ CP2 whose image is C.

Here, generically injective means birational (←→ isomorphic on an open set).

Proof: If F is irreducible then C[x0, x1, x2]/(F, x2 − 1) is an integral domain and has a quotient fieldK. The mapping f(P ) = (x0(P ) : x1(P ) : 1) extends to by continuity to a desingularization of C.

Theorem 2.8.4 (Hurwitz). If f : Z1 −→ Z2 is a finite map, then KZ1 ∼ f∗KZ2 +R (f∗ = pullback) whereR is the ramification divisor

∑p∈Z1

rp(f) · p, rp(f) = ordp(f)− 1, where f : Z1 −→ Z2 is locally at p of the

form f(z) = zordp(f) and

deg(f) =∑

p∈ fibre

ordp(f),

where this sum is independent of the choice of the fibre.

Corollary 2.8.2 (Riemann-Hurwitz).

deg(KZ1) = (deg(f)) · (deg(KZ2

)) + deg(R)

Corollary 2.8.3. g(Z1) ≥ g(Z2), where g is for genus.

22

From this it follows that there exists no map from a rational curve to an elliptic curve.

The following equality is the algebraic geometry definition of topological genus:

g(Z) = dim(Γ(K2)) = dimC(Hol′(Z))

where Hol′(Z) is the set of holomorphic differentials on Z.

23

2.9 Topology of Riemann surfaces

One known fact about Riemann surfaces is that any Riemann surface is orientable. Recall that a manifoldis orientable if its transition functions have positive Jacobian . Let Z be a Riemann surface, consider twointersecting charts Uα and Uβ , and let z ∈ Uα ∩ Uβ . Then we have

df

dz(z) = f ′(z) as an R-matrix.

We write f = u+ iv, then df = α+ iβ.

C

αβα β−1

holomorphic

By the Cauchy-Riemann equations, we have

df =

(α β−β α

)

and so

Jac(α β−1) = det(D(α β−1)) = |df |2 = det(df) = α2 + β2 > 0.

It is also known that every Riemann surface is obtained by attaching handles to CP1 = S2.

# handles = # holes = topological genus = g:

∞

0

Theorem 2.9.1. Any Riemann surface is triangularizable.

24

This fact is easy to prove for Compact Riemann surfaces, and shows that any such is obtained by attachinghandles to S2 = CP1 = C ∪ ∞. We denote Zg for a Riemann surface Z of genus g. It is known that thefirst homotopy group of Zg is given by

π1(Zg) = Za1 ∗ Zb1 ∗ · · · ∗ Zag ∗ Zbg/⟨a1b1a

−11 b−1

1 · · · agbga−1g b−1

g

⟩

Notice that every hole gives two generators.

a1

b1bg

ag

b−1g

b−11

a−11

a1

b1

Let Ai denote the space of C∞-complex valued i-forms on a Riemann surface, then

A = A1,0 ⊕A0,1,

ω′ = fdx+ gdy = hdz + hdz,

where hdz ∈ A1,0 and hdz ∈ A0,1. We also have differential operators making the following diagram commutes

A1

C∞ − functions = A0 A0,1 A2

A1,0

d

∂

∂

d

∂

∂

where d = ∂ + ∂.

Definition 2.9.1. An i-form ω ∈ Ai is called closed if dω = 0. It is called exact if ω = dα.

Since d d = 0, we have Im(d) ⊆ Ker(d), and so we define the de Rham cohomology groups of Z and thequotient

HidR(Z) = closed i-forms/exact i-forms

25

Note that H0dR(Z) = C if Z is connected. If Z is also compact, by the Poincare duality Theorem we have an

isomorphism H2dR(Z) −→ H0

dR(Z) given by

[ω] 7→∫

Z

ω

Also, HndR(Z) = 0 for every n ≥ 3. For any compact and connected Riemann surface Zg, the middle

cohomology is given by

H1dR(Zg) = π1(Zg)/ 〈commutator subgroup〉 = Za1 ⊕ Zb1 ⊕ · · · ⊕ Zag ⊕ Zbg

Theorem 2.9.2. Let ω be a holomorphic differential (←→ 1-form). Then ω is d-closed (hence [ω] ∈ H1dR(Z)).

Proof: Locally, ω = f(z)dz where f ∈ O. Then

dω = (∂ + ∂)ω = ∂ω + ∂ω =

(∂f

∂zdz

)∧ dz +

(∂f

∂zdz

)∧ dz

= ∂f ∧ dz + ∂f ∧ d = 0 + 0dz ∧ dz, since f is holomorphic.

26

2.10 Product structures on⊕

iHidR(Z)

By taking wedge products of forms, we have a surjective map H1dR(Z)×H1

dR(Z) −→ C given by

([ω1], [ω2]) 7→∫

Z

ω1 ∧ ω2

In this situation we have H1dR(Z) ∼= (H1

dR(Z))∨ (finite dimesnion) and so the previous map is a perfect pairing.

We also have the cap product ∩ : H1(Z)×H1(Z) −→ Z which gives rive to a perfect pairing.

By the Poincare duality Theorem we have H1dR(Z) ∼= H1(Z)∨ ⊗ C. Also

H1dR(Z) = (H1(Z)∗ = H1(Z))⊗ C.

Recall that the Euler characteristic of Z is given by

χ(Z) :=∑i(−1)idimHi

dR(Z) =∑i(−1)idimZHi(Z) = 2− 2g

Recall H0dR(Z) = H2

dR(Z) = C and H1dR(Z) = C2g.

Definition 2.10.1. Let Ω be the space of differentials (= holomorphic 1-forms) on a compact Riemannsurface Z. The geometric genus of Z is defined by

Pg(Z) = dimC(Ω)

Clearly, Ω → H1dR(Z) if Z is compact. We have that

fdz =dg

dzdz = dg = 0

implies that g ∈ O. Since Z is compact, we get that g = const. Similarly, Ω → H1dR(Z). It follows 2Pg ≥ 2g.

It is difficult to determine when they are equal. The fact that Pg ≥ 2 implies that there exist meromorphicfunctions. If Pg = g, then a loop C on a compact Riemann surface Z is homotopic to 0 if and only if

∫Cω = 0,

for every ω ∈ Ω. The map

ω 7→∫Cω

is called the period map.

Definition 2.10.2. Let S be a Riemann surface, define

H1(S;Z) = π1(S)/[π1(S), π1(S)]

where [π1(S), π1(S)] denotes the commutator subgroup of π1(S).

Note that if S is compact then H1(−) is a free Z-module generated by a1, b1, . . . , an, bn. We can considereach ai and bi as maps S1 −→ S or curves with parameter t ∈ [0, 1] in S, oriented counterclockwise.

27

a1

b1bg

ag

Since S is oriented we have that there exists an intersection pairing in H1 (any two loops, or elements in H1,are homotopic to loops which are transversal). Note that a and b are the same element in H1 if and onlyif a + (−b) = a − b = ∂U , for some open set U ⊆ S, i.e., a is homologous to b. Here −b denotes reverseorientation. So if a is homotopic to b then they are homologous.

Consider the following picture:

p −→a

−→b y

x

C

chart

We have −→a ∧p−→b = kx ∧p y, and denote

(a, b)p = sign(k)

and(a, b) =

∑p∈a∩b(a, b)p

Since the previous sum does not depend on the choice of the prepresentative, we have a well defined map

( , ) : H1 ×H1 −→ Z

If S is a Riemann surface of genus g, we have in terms of a basis that

(ai, bj) = (bi, bj) = 0, for every i, j,

(ai, bj) = −(bj , ai) = δij .

28

Then we have that each (ai, bj) is a skew-linear pairing which is unimodular (det = 1).

0 1−1 0

1det

ai bi

ai

bi

Hence H1(S;Z) = Hom(H1(S;Z),Z) =: H1(S;Z).

Let ω ∈ M′(S) be a closed meromorphic differential. Recall that ω is exact if and only if ω = df for somemeromorphic function f ∈M(S). Can we choose a function f ∈M(S) such that f =

∫ zpω? As an exercise,

think if it is possible that Resp(ω) = 0 for every p ∈ S. Consider the period homomorphism

Πω : H1(S;Z) −→ C

[γ] 7→∫

γ

ω, γ is a loop.

This map is well defined. For if γ = γ′ in H1 then γ − γ′ = ∂u, and by the Stokes Theorem we have∫γω =

∫γ′ω. Also, Πω is a C-linear map.

By the de Rham Theorem, we have an isomorphism H1dR(Z) ∼= H1(S;Z) ⊗ C if Z is compact. Such an

isomorphism is given by[ω] ∈ H1

dR(Z) 7→ Πω

Corollary 2.10.1. dimCH1dR(S) = 2g.

Proof: We have

2− 2g = χ(S)

=∑

(−1)ihidR(S), where hidR(S) = dim(H1dR(S)),

= h0dR(S)− h1

dR(S) + h2dR(S)

= 1− dimCH1dR(S) + 1.

Theorem 2.10.1. The de Rham ismomorphism carries wedge product of forms defined by

H1dR ×H1

dR −→ C

([ω1], [ω2]) 7→∫

S

ω1 ∧ ω2

isomorphically to the (cup) product on H1(S).

29

Recall that Ω denotes the space of holomorphic 1-forms on S. Since

H1dR

∼=−→ H1(S;Z)⊗ C = HomC(H1,C)

if S is compact, we have an inclusion Ω → H1dR if S is compact. We have

ω exact =⇒ ω = df =⇒ f holomorphic =⇒ f constant =⇒ ω = 0

Definition 2.10.3. Pg = dim(Ω) (geometric genus).

It is known that Pg ≥ g. When the equality holds, it is because of the Hodge Decomposition Theorem.

30

2.11 Questions about (compact) Riemann surfaces

Recall that if f : Z1 −→ Z2 is a finite map of degree n of Riemann surfaces, then any meromorphic functionon Z1 satisfies a polynomial of degree n over the field of Z2. Hence

f∗M(Z2) → f∗(Z1)

is a field extension of degree 6= n.

Fact 2.11.1. The equality deg = n implies that all finite extensions ofM(Z2) are in natural bijective corre-spondence with finite maps up to isomorphisms, and each extension is Galois if and only if so is the finite map.

Example 2.11.1. If Z2 = P1, M(P1) = C(Z) then there is a bijective correspondence between fields oftranscendence degree = 1 and finite covering of P1.

The equality deg = n implies that Aut(Z) = C-Aut of M(Z).

Question: We know that 2Pg ≥ 2g. When are they equal?

If yes, then a loop C is homologous to 0 if and only if∫Cω = 0 for every ω ∈ Ω. Also, H1

dR = Ω ⊕ Ω. Thevalues

∫aiω and

∫biω are called periods.

31

2.12 Harmonic differentials and Hodge decompositions

Recall that A1 = C∞-valued 1-forms.

Definition 2.12.1. Notice that locally every C∞-valued 1-form can be written as ω = fdz + gdz, wherefdz ∈ A1,0 and gdz ∈ A0,1. There exists a C-linear map ∗ : A1 −→ A1 called the star operator, locallydefined by

∗(pdx+ qdy) = −qdx+ pdy

Or equivalently,

∗(fdz + gdz) = i(−fdz + gdz).

Every ω is uniquely a sum of ω1,0 ∈ A1,0 and ω0,1 ∈ A0,1, and

∗(ω) = i(−ω1,0 + ω0,1).

Definition 2.12.2. ω ∈ A1 is harmonic if dω = 0 = d(∗ω). An 1-form ω is called coclosed if d(∗ω) = 0.In other words, ω is harmonic if it is closed and coclosed.

Locally, ω = fdz + gdz is closed if gz = fz, and is coclosed if gz = −fz. Then if ω is harmonic we havegz = fz = 0, i.e, ω = fdz + gdz ∈ Ω⊕Ω, where fdz is a holomorphic 1-form and gdz is an anti-holomorphic1-form.

Proposition 2.12.1. Let H1 be the space of harmonic differentials on Z. Then H1 = Ω⊕ Ω.

The Hodge Theorem states that H1 = H1dR.

Example 2.12.1. Ω is nonempty for C/(Z⊕ Z).

We have a positive answer to all questions we established.

Theorem 2.12.1 (Hodge Decomposition). H1dR(Z) = H1 = Ω⊕Ω, where the first equality is known as the

Hodge Theorem.

Theorem 2.12.2 (Riemann Existence Theorem). Let Z be a local coordinate around p ∈ Z, and n ≥ 1.Then there exists an exact harmonic differential ω on Z − p such that

ω − d(

1

zn

)= ω +

n

zn+1dz

is harmonic on a neighbourhood U of p, and ω ∈ B′S−U , i.e.,

∫S−U ω ∧ ω <∞.

32

Corollary 2.12.1.

(1) There exist meromorphic differentials on Z with any preassigned finite set of poles p and any principalparts

ωp =

∞∑

i=−naiz

idz, when n ≥ 2.

(2) There exist a meromorphic function with any prescribed value at a finite set of points.

(3) f∗M2 →M1 has degree = deg(f) for a finite map f : Z1 −→ Z2.

33

2.13 Analysis on the Hilberts space of differentials

There exists a Hermitian inner product for 1-forms ω1 and ω2 (at least one which is compactly supported)

(ω1, ω2) =∫Zω1 ∧ ∗ω2 <∞ .

Locally, ωi = pidx+ gidy, with i = 1, 2. Then

ω1 ∧ ∗ω2 = (p1p2 + g1g2)dx ∧ dy.

Hence we can define

||ω||2L2 =

∫

Z

ω ∧ ∗ω <∞.

Definition 2.13.1. Let B′ be the space of bounded 1-forms ω such that ||ω||2 <∞.

With respect to the L2-norm, B1 is a Hilbert space.

Let E be the closure in B1 of dA0C , where A0

C is the space of C∞-functions with compact support.

Theorem 2.13.1 (Orthogonal decomposition). Let ω ∈ B1. Then there exists a unique orthogonal decom-position

ω = ωh + df + ∗dgwhere ωh is bounded harmonic and f, g ∈ A0, and df, dg ∈ E.

Proof: The essential point is that the space H is orthogonal to both E and ∗E, and E ⊥ ∗E. Ifψ,ϕ ∈ A0 then

⟨dϕ, ∗dψ

⟩= −

∫

Z

dϕ ∧ dψ =

∫

Z

ψddϕ+

∫

Z

d(ψdϕ)

= 0 + 0.

Similarly, saying that ω is closed means that it is orthogonal to ∗E, and coclosed means that it isorthogonal to E. For example,

0 = 〈d ∗ ω, ϕ〉 =

∫d ∗ ω ∧ ϕ =

∫dϕ ∧ ∗ω = 〈dϕ, ω〉

= 〈ω, dϕ〉 .

HenceH⊕⊥ E ⊕⊥ ∗E → B1.

To show the equality, we go to the L2-completion of B′ first.

Theorem 2.13.2 (Regularity). H = (E ⊕ ∗E)⊥ in B1 where (∨) means L2-completion.

34

Corollary 2.13.1 (Hodge decomposition). H1dR(Z) = Ω⊕ Ω.

Proof: Saying that a form is closed means that it is orthogonal to ∗E. Hence the previous theoremimplies that a closed 1-form ω is uniquely ωh + df , i.e., every element of H1

dR has a unique harmonicrepresentative.

35

2.14 Review

Theorem 2.14.1 (Orthogonal decomposition). For every ω ∈ A1 = space of differntial 1-forms, there existsa unique decomposition

ω = ωh + df + ∗dgwhere ωh is a bounded harmonic, f, g ∈ A0 = space of smooth functions.

We denote A = the space of C∞-functions.

Proof: Let H be the space of harmonic differentials. Then H is orthogonal to both E = dA0 and∗E = ∗dA0. This fact follows easily using the L2-inner product and the equality ∗∗ = (−1)k, forRiemann surfaces one has (−1)k = 1. We have

H⊥E⊥ ∗ E ⊆ A1.

Taking completion, we have

H⊥©E⊕∗E = A1

where H⊥©E is the orthogonal (H ⊥ E) direct sum of H and E. By the Weyl’s Lemma, we have H = H.

Lemma 2.14.1. Any distribution (1-form) T (1-current) with ∆T = 0 is the distribution of some differentialfunction f , i.e., T = Tf where

Tf [h] =

∫

U

hf

and h is compactly supported in U ⊂⊂ Z, where Z is a Riemann surface.

Corollary 2.14.1 (Hodge decomposition). H1dR(Z) = Ω⊕ Ω = H.

36

2.15 Proof of Weyl’s Lemma

Claim 2.15.1. If f ∈ A0(C) then there exists ψ ∈ A0(C) such that ∆ψ = f .

Definition 2.15.1. A0c(U) = f ∈ C∞(U) / f has compact support in U.

Note that A0c(U) is a topological space of uniform convergence (not complete).

A distribution on U is a continuous linear functional T : A0c(U) −→ C.

Example 2.15.1. Let h ∈ A0c . Define

Th[f ] =

∫

U

hf

Then T is a distribution. Using integration by parts, for h ∈ A0c and f ∈ A0

c , we have∫hDαf = (−1)|α|

∫

Z

fDαh,

where α = (α1, α2), Dα = ∂(α1+α2)

∂xα1∂yα2and |α| = α1 + α2. In other words, we have

TDαh[f ] = (−1)|α|Th[Dαf ]

Definition 2.15.2. (DαT )[f ] = (−1)|α|T [Dαf ], for every f ∈ A0c .

The definition for DαTh is the same as TDαh. Let g ∈ A0(U × I) where I is an interval, supp(g) ⊆ K × Iand K ⊂⊂ U .

Let fε(z) = g(z,t+ε)−g(z,t)ε . Then fε −→ ∂g

∂t as ε −→ 0. By continuity of T , we get

d

dtTz(g(z, t))

ε→0←− T [fε]ε→0−→ Tz

[∂g(z, t)

∂t

]

So:

Claim 2.15.2. ddtTz[g(z, t)] = Tz

[∂g(z,t)∂t

].

Similarly, if U and V are open in C and K ⊂⊂ U , L ⊂⊂ V , g ∈ A0(U × V ) are such that supp(g) ⊆ K × L,then

Tz

[∫

V

g(z, ε)d(Volε)

]=

∫

V

Tz(g(z, ε))dε

Let ρ ∈ A0(C) satisfy supp(ρ) ⊆ D, ρ(z) = ρ0(|z|) for every z, and∫C ρ = 1. Set ρε(z) = 1

ε2 ρ(zε

). Then

f ∈ A0(U) implies

(ρε ∗ f)(z) =

∫

U

ρε(z − ε)f(ε)dVol(ε) ∈ A0(U (ε))

where U (ε) = z ∈ U / d(z, ∂U) > ε.

37

Claim 2.15.3. For every α ∈ N2, we have

Dα(ρε ∗ ρ) = ρε ∗ (Dαf)

This result follows similarly applying a change of variables z 7→ z + ε.

Claim 2.15.4. If z ∈ U (ε) and f is harmonic on D(z, ε), then

(ρε ∗ f)(z) = f(z).

Proof:

(ρε ∗ f)(z) =

∫

D(0,z)

ρε(ε)f(z + ε)dVol(ε) =

∫ 2π

0

∫ ε

0

ρε(r)f(z + rεiθ)rdrdθ = 2πf(z)

∫ ε

0

ρε(r)rdr = f(z).

Proof of Weyl’s Lemma: Since ε −→ ρε(ε− z) has compact support in U , for every z ∈ U (ε),

h(z) := Tε[ρε(ε− z)]

is defined and belongs to A0(U (ε)) by Claim 6.2. By Claim 6.4, it suffices to show that for everyf ∈ A0

c(U(ε)) we have

T [f ] =

∫

U(ε)

hf,

where h is harmonic since∆h = Tε[∆zρε(ε− z)] = ∆Tε[ρ(ε− z)] = 0.

We want to show that T = Th. We have

T [ρε ∗ f ] =

∫

U(ε)

hf.

Hence it suffices to showT [f ] = T [ρε ∗ f ].

By Claim 6.1, there exists ψ ∈ A0 such that ∆ψ = f , where ψ is harmonic on V = C− supp(f). Henceψ = ρε ∗ ψ on Vε by Claim 6.4. Therefore, O = ψ − ρε ∗ ρ ∈ A0

c(U) satisfies

∆O = ∆ψ − ρε ∗∆ψ = f − ρ ∗ f

since ∆T = 0, and T (∆ψ) = 0. Hence

T [f ] = T [ρ ∗ f + ∆O] = T [ρε ∗ f ].

38

2.16 Riemann Extension Theorem and Dirichlet Principle

Theorem 2.16.1 (Riemann Extension Theorem). Let Z be a Riemann surface and p ∈ Z. For n ≥ 1 thereexists a harmonic differntial ω on Z − p such that:

(1) ω − d(

1zn

)is harmonic on a small neighbourhood of p.

(2) ω ∈ B′Z−U , i.e., ω is bounded and smooth outside U , with ||ω||2L2 <∞.

Outline of the proof: Let ρ(z) be a differentiable function on Z such that ρ ≡ 0 outside U and ρ ≡ 1on a neighbourhood of p. The form

ψ = d

(ρ(z)

zn

)∈M(p).

We take p = 0. Now ψ− i ∗ (ψ) is smooth and has compact support on Z (≡ 0 on a neighbourhood of 0and outside U), where ∗(ψ) = i(udz − vdz) if ψ = udz + vdz. Hence by the Decomposition Theorem wehave

ψ − i ∗ (ψ) = ωh + df + ∗(dg)

andω = ψ − df = ωh + i ∗ (ψ) + ∗(dg)

It follows that ω is harmonic ((d ∗ d)ω = 0) since ψ and df are exact.

The uniqueness of ω can be guaranteed by adding

(3) (ω, dh) = 0 for every dh ∈ A1 such that ||dh|| <∞ and dh ≡ 0 on a neighbourhood of p.

This condition is the same as the following: Since dh ≡ 0 on a neighbourhood N of p, we have

||ω + dh||2Z−N = ||ω||2Z−N + (dh, dh) + (ω, dh) + (ω, dh)

= ||ω||2Z−N + ||dh||2Z−N≥ ||ω||2

Z−N .

The Dirichlet Principle states that harmonics ω minimizing || ||2Z−N is given by the class of all differentials

ω + dh such that dω = 0 on N . So, uniqueness is an easy consequence of this.

39

2.17 Projective model

Theorem 2.17.1. Any compact Riemann surface can be embedded in a projective space.

Proof: f = (1, f1, f2, . . . , fn) : Z −→ CPn if f(P ) = f(Q) for P 6= Q. Then just adding a meromorphicfunction that separates P and Q. This process must terminate by compactness.

Theorem 2.17.2 (Chow). The image of f is a projective algebraic variety, i.e., it is V (P1, . . . , PL), forsome polynomials P1, . . . , Pl on Pn.

Proof: Case n = 2: Consider f : Z −→ CP2 and [z0, z1, z2] ∈ C3. Then a = f∗(z0z2

)and b = f∗

(z1z2

)

are algebraic dependent over C, i.e., there exists a polynomial F (Z1, Z2) of degree d such that F (a, b) = 0.

Then xd2F(x0

x2, x1

x2

)defines f(Z).

This result is also known as the 1-st GAGA Principle (after Geometrie Algebrique et Geometrie Analytique).

40

2.18 Arithmetic nature

Recall that an (algebraic) number field F is a finite degree (and hence algebraic) field extension of Q.

An arithmetic Riemann surface is a Riemann surface contained in Pn defined over a number field (i.e.,(P1, . . . , Pn) = 0 defines a Riemann surface and P1, . . . , Pn have coefficients in the same number field).

Theorem 2.18.1 (Belgi ’79). A Riemann surface Z is arithmetic if and only if there exists a holomorphicmap Z −→ CP1 with 3 ramification points.

Theorem 2.18.2 (Mardell and Faltings). A Riemann surface Z defined over a number field has a finitenumber of rational points, i.e., if g(Z) ≥ 2.

Classification: π1(Z) = 0 if and only if the Riemann Mapping Theorem holds, Z ∼= CP1,C or D.

Corollary 2.18.1. A Riemann surface Z is rational (∼= CP1) if and only if g(Z) = 0.

41

42

Chapter 3

COMPLEX MANIFOLDS

3.1 Complex manifolds and forms

Recall that for a smooth R-manifold M , there is an ideal I(x) for each x ∈M , given by

I(x) = f ∈ C∞(M) / f(x) = 0 → C∞(M)

The cotangent plane at x ∈M can be defined as the quotient

T∨x (M) := I(x)/I(x)2

and the tangent plane at x ∈M is simply the dual space of the cotangent plane T∨x (X), i.e.,

Tx(M) := (T∨x (M))∨

Definition 3.1.1. An almost complex structure on an R-differentiable manifold X of dimR = 2n is anepimorphism J of TX such that J2 = −1. Or equivalently, it is the structure of a complex vector bundle onTX.

A complex structure on X induces an almost complex structure on X by setting J = i =√−1. We obtain a

map J : TX,R −→ TX,R with√−1 acting on the domain and J acting in the codomain. We have

∂

∂z=

1

2

(∂

∂xi− i ∂

∂yi

)7→(

∂

∂xi,∂

∂yi

)

Locally, J is defined by (∂

∂xi,∂

∂yi

)7→(∂

∂yi,− ∂

∂xi

)

Let (X,J) be an almost complex manifold. Then TX,R ⊗ C contains an eigen-bundle T 1,0X corresponding to

the eigen-value i, an an eigen-value T 0,1X corresponding to the eigen-value −i, for the operator J . Note that

T 1,0X is naturally isomorphic to TX,R by taking the real part, and this isomorphism identifies i with J . Hence

T 1,0X is generated by vectors of the form u− iJu, with u ∈ TX,R.

Theorem 3.1.1. A complex manifold has a complex structure J on TX,R and its associated subbundle

T 1,0X ⊆ TX,R ⊗ C is naturally the same as TX (by taking the real part).

43

Similarly, if ΩX,R := T∨X,R, then

ΩX,R ⊗ C = ΩX ⊕ ΩX

with the identification

ΩX ←→ Ω1,0X ←→ dzi,

ΩX ←→ Ω0,1X ←→ dzi.

Definition 3.1.2. An almost complex structure is called integrable if it comes from a complex structure.

The only spheres with an almost complex structure are S2 and S6. The 6-sphere S6 has an almost com-plex structure via the octonions, by taking the multiplication structure in the multiplicative structure in thesphere in the purely imaginary part of octonions.

Question: Does it exist complex structures on S2?

Theorem 3.1.2 (Newlande - Niremberg). J is integrable if and only if [T 1,0X , T 1,0

X ] ⊆ T 1,0X .

Proposition 3.1.1 (Poincare Lemma). Let α be a closed differential from a differentiable manifold withdeg(α) > 0. Then locally α = dβ, for some form β.

Proposition 3.1.2 (∂-Poincare Lemma). Let α be a ∂-closed differential from a differentiable manifoldwith (p, q) = deg(α) and q > 0. Then locally α = ∂β, for some form β.

Proof: First we show that we can reduce the problem to the case p = 0 and q = 1. In general, we know

α =∑

αI,JdzI ∧ dzJ

∂α =∑

dαI,J ∧ dzI ∧ dzJ = 0

Then αI =∑αI,J

∂zJ is ∂-closed and of type (0, q), and so αI = ∂β if and only if α = (−1)p∂(∑∂zI∧βI).

Henceforth assume that α is of type (0, q), i.e., α =∑αJdz

J . Apply the induction on the largestinteger k such that k ∈ J and αJ 6= 0. Necessarily k ≥ q and k = q implies α = fdz1 ∧ · · · ∧ dzq. In thelatter case ∂α = 0 if and only if f is holomorphic in the variables zi with i > q. We may know applythe following result:

Proposition: There exists a differentiable function g holomorphic in the variables zi, with i > q, suchthat ∂g

∂z q= f and hence α = (−1)q−1∂(gdz1 ∧ · · · ∧ dzq−1).

Now assume the ∂-Poincare Lemma proved for k− 1 > q. Write α = α1 +α2dzk, α2 =∑α2,Jdz

J where|J | = q − 1 and J ⊆ 1, . . . , k − 1. So ∂ = 0 implies α2,J is holomorphic in variables zl, with l > k.

Hence by the previous proposition, we have α2,J =∂β2,J

∂zk, where β2,J is holomorphic in zl, l > k. Then

∂β = ∂(β2,JdzJ) = (−1)q−1α2 ∧ dzk + α′1

44

where α′1 involves only the coordinate zl for l < k. Thus,

α = α′′1 + ∂β

where α′′1 for l < k. Since β is holomorphic in the zl for l > k, we have ∂α′′1 = 0, q = deg(α′′1) < k. Weconclude by induction.

Proof of the previous proposition: We restrict to the case p = 0 and q = 1. Let α = fdz. Asstatement is local, we may assume that supp(f) is compact. Define

g =1

2πi

∫

C

f(z)

ζ − z dζ ∧ dζ := limε→0

1

2πi

∫

C\D(z,ε)

f(z)

ε− z dz ∧ dz.

This limit exists since f is bounded and 1ζ−z in integrable on D. We want to show that ∂g = α = fdz.

Now g = gε where gε = 12πi

∫C\D(0,ε)

f(γ+z)γ dγ ∧ dγ. Then

∂gε(z) =1

2πi

(∫

C\D(0,ε)

∂zf(γ + z)dγ ∧ dγ

γ

)dz

implies

∂

∂zg(z) = ∂zg(z) =

1

2πi

∫

C

∂

∂zf(γ + z)

dγ ∧ dγγ

= limε→0

1

2πi

∫

C\D(z,ε)

∂

∂ζf(ζ)

dζ ∧ dζζ − z .

On C\D(z, ε), we have ∂f

∂ζ(z)dζ∧dζζ−z = −dε

(f(ζ) dζ

ζ−z

). By the Stokes Theorem, we get

1

2πi

∫

C\D(z,ε)

∂f

∂ζ

dζ ∧ dζζ − z =

1

2πi

∫

∂D(z,ε)

f(ζ)

ζ − z −→ f(z) as ε −→ 0.

Hence ∂g = fdz.

45

3.2 Kahler manifolds

Let V be a complex vector space with J =√−1 and W = HomR(V,R). Recall VC := V ⊗ C = V 1,0 ⊕ V 0,1.

Hence also WC = W 1,0 ⊕W 0,1 ⊇W .

Definition 3.2.1. Let W 1,1 = W 1,0 ⊗W 0,1 ⊆ ∆2WC ⊇ ∆2W ,

W 1,1 = (1, 1)-forms = sesqui-linear forms on V .

Let W 1,1R = W 1,1 ∩ ∆2W = real (1, 1)-forms = real 2-forms of type (1, 1) = alternating forms. A

(1, 1)-form h ∈ W 1,1 is called Hermitian if h(u, v) = h(v, u) for every u, v ∈ V . Let W 1,1H be the space of

such forms.

Fact 3.2.1. There exists a bijective correspondence between Hermitian forms and real alternating forms oftype (1, 1) via

W 1,1H 3 h←→ Im(h) ∈W 1,1

R

Proof: Since h(u, v) = h(v, u), we have that Im(h) is alternating on V , i.e., Im(h) ∈ ∆2W . Conversely,let ω ∈W 1,1

R and set

g(u, v) = ω(u, Jv) = −ω(Ju, v) and

h(u, v) = g(u, v)− iω(u, v).

Then g(u, v) = g(v, u) and thus h(u, v) = h(v, u), i.e., h is Hermitian.

Locally, ω =∑

i2aijdzi ∧ dzj = −Im(h) ∈ Ω1,1

X ∩ Ω2X,R, where (aij) is hermitian.

Definition 3.2.2. ω ∈W 1,1R is positive if the correspondence h is positive definite.

Definition 3.2.3. A positive real (1, 1)-form on an almost complex manifold (X, J) is a C∞ associated ofa positive real (1, 1)-form on each tangent space TX,x, x ∈ X.

Definition 3.2.4. A Hermitian metric on a complex vector bundle E over a smooth manifold M is anelement h ∈ Γ(E ⊗ E)∗. A Hermitian manifold is a complex manifold with a Hermitian metric on itsholomorphic tangent space. Likewise, an almost Hermitian manifold is an almost complex manifold witha Hermitian metric on its holomorphic tangent space.

Corollary 3.2.1. There exists a bijective correspondence between real (1, 1)-forms ω on a complex manifoldM and Hermitian metrics on M .

Definition 3.2.5. Let h be a Hermitian metric. We shall say that h is Kahler if ω = Im(h) is closed.

46

Corollary 3.2.2. If a symplectic structure ω on a complex manifold is positive of type (1, 1) (i.e., it vanisheson Ω2,0 and hence also on Ω0,2 and its associated h is positive definite), then it is −Im(h) for a Kahler metric.

Corollary 3.2.3. A Hermitian metric can always be written as h = g + iω where g is a Riemannian metricinvariant under J (g(u, v) = g(Ju, Ju)) and ω is a positive (1, 1)-form, g(u, v) = ω(u, Jv).

Definition 3.2.6. A pair (X,ω) formed by a complex manifold X and a positive (1, 1)-form ω is called aKahler manifold.

Lemma 3.2.1. dVolh = ωn

n! for (X,h), h = hω = g(u, v), where ωn = ω ∧ · · · ∧ ω of type (n, n).

Proof: Let ei be an orthonormal basis of TX,x with respect to h. Then ei, Jei is a real orthonormalbasis for TR

X,x with respect to g with positive orientation. It suffices to check ωn

n! = dx1∧dy1∧· · ·∧dxn∧dynwhere dx1, dy1, . . . , dxn, dyn is the dual basis to (ei, Jei). Let dzj = dxj + dyj . Then we have ωx =i2

∑j dzj ∧ dzj and

ωnxn!

=

(i

2

)n∏

j

dzj ∧ dzj at x,

i

2dzj ∧ dzj = dxi ∧ dyj .

Corollary 3.2.4. If X(n) is a compact Kahler manifold then [ωk] ∈ H2kdR(X) is nonzero for every k < n.

Proof: ωk = dγ implies ωn = d(ωn−k ∧ γ). The last implies 0 6=∫Xωn = 0, getting a contradiction.

Corollary 3.2.5. Let X(k) be a compact Kahler submanifold M . Then [x] ∈ H2k(X) is nonzero.

Proof: Clearly hM |TX = hX and i∗ω(M,h) = ω(X,hX), where i : X →M is the inclusion. If i(X) = ∂Γ,then by the Stokes Theorem

VolumeX =

∫

X

i∗ωhM =

∫

Γ

dωhM = 0 since dωhM ≡ 0.

47

3.3 Metrics and connections

Let E −→ X be a C∞-vector bundle on X, and let Ai(E) be the vector space of C∞ E-valued forms on X.

Definition 3.3.1. A real (complex) connection on E is a real (resp. complex) linear map

∇ : A0(E) −→ A1(E)

satisfying the Leibniz rule:∇(fσ) = df ⊗ σ + f∇σ.

For a vector field ψ and σ ∈ A0(E) we write

∇ψσ = (∇σ)(ψ) ∈ A0(E).

In the case where E is a holomorphic vector bundle, we have the operation

∂E : A0(E) −→ A0,1(E)

which defines holomorphic sections of E via Ker(∂E). It satisfies the ∂-Leibniz rule instead:

∂(fσ) = ∂f ⊗ σ + f∂σ

but it is not a complex connection.

Proposition 3.3.1 (For a Riemannian manifold). If (M, g) is a R-manifold then there exists a uniqueconnection ∇ on TM called the Levi-Civita connection satisfying:

(1) d(g(ψ1, ψ2)) = g(ψ1,∇ψ2) + g(∇ψ1, ψ2), i.e., g is ∇-invariant.

(2) ∇ψ1ψ2 −∇ψ2ψ1 = [ψ1, ψ2], i.e., g is torsion free of ∇.

Theorem 3.3.1 (and definition). Let E −→ X be a holomorphic vector bundle with a Hermitian metric.There exists a unique complex connection ∇ on E, called the Chern connection satisfying:

(1) d(h(σ, τ)) = h(∇σ, τ) + h(σ,∇τ), i.e., ∇ is invariant under (or compatible with) h.

(2) Let ∇0,1 be its composition with A1(E) −→ A0,1(E). Then ∇0,1 = ∂E .

Theorem 3.3.2. The following statements for a complex Hermitian manifold (X,h) are equivalent:

(1) h is Kahler.

(2) J is flat for the Levi-Civita connection.

(3) Chern connection = Levi-Civita connection.

48

3.4 Review

A complex structure on a real manifold M of dimension 2n is an endomorphism J of TM such that J2 = −1.If M is complex, normally we take J =

√−1. A real 2-form h is Hermitian if h(u, v) = h(v, u). It is know

that a form h is Hermitian if and only if h is a positive (1, 1)-form.

Theorem 3.4.1. There exists a bijective correspondence between real alternating forms of type (1, 1) andHermitian metrics. Such a correspondence is given by

W 1,1H

∼−→W 1,1R

h 7→ Im(h)

where Im(h) is a symplectic 2-form.

Theorem 3.4.2. The following conditions are equivalent for a complex Hermitian manifold (X,h):

(i) h is a Kahler metric, i.e., dwh = 0.

(ii) J is flat for the Levi-Civita connection of h.

(iii) The Chern connection of h on T 1,0M equals the Levi-Civita connection on TR

M .

Proof:

• (iii) =⇒ (ii): It is clear because the Chern connection is C-linear by definition.

• (ii) =⇒ (i): Condition (ii) means that the Levi-Civita connection commutes with J . Then

dω(ϕ1, ϕ2) = ω(∇ϕ1, ϕ2) + ω(ϕ1,∇ϕ2).

Let C∞(M) 3 ϕ[ω(ϕ1, ϕ2)] = ω(∇ϕϕ1, ϕ2) + ω(ϕ1,∇ϕϕ2). Since

dω(ϕ,ϕ1, ϕ2) = ϕω(ϕ1, ϕ2)− ϕ1ω(ϕ,ϕ2) + ϕ2ω(ϕ1, ϕ)− ω([ϕ,ϕ1], ϕ2)

the result follows from [ϕi, ϕj ] = ∇ϕ1ϕj −∇ϕ2

ϕ1.

• (i) =⇒ (iii): The Chern connction equals the Levi-Civita connection for the flat metric∑i dzi∧dzi.

The result follows from the following proposition.

Proposition 3.4.1. If (X,h) is a Kahler manifold and if x ∈ X, then there exists a holomorphic coordinate(z1, . . . , zn) centred at x such that

hij = h

(∂

∂zi,∂

∂zj

)= Im +O(

∑|zi|2).

The converse is also true.

49

3.5 The Fubini Study metric

Let L = (l, v) ∈ CPn × Cn / v ∈ l. Consider the diagram

L CPn × Cn+1 Cn+1

CPn

i π2

π1π1 i

The composite map π2 i is called the blow up at 0. We have that L is a holomorphic line bundle over CPnand is denoted by O(−1).

Definition 3.5.1. OCPn(h) := L−k where L−k := (L∨)⊗k for k > 0, is a holomorphic line bundle over CP1.

The standard metric∑ |zi|2 on CPn+1 restricts to a Hermitian metric on L. Its curvature (Ricci or Chern

form) is given by

ω = σ∗2

2π∂∂log|zi|i =

i

2π∂∂log|σ|2

for any choice of a holomorphic section σ of L over CPn. Therefore, σ′ = σf , for f ∈ O and so

log(σ′)2 = log|σ|2 + log|f |2,

where log|f |2 = logf + logf and it is ∂∂-closed.

Lemma 3.5.1. ω is a positive (1, 1)-form.

Proof: We prove only the case n = 1. We have

∂log(1 + |z|2) =∂(1 + |z|2)

1 + |z|2 =zdz

1 + |z|2 .

So

ω =i

2π

[(1 + |z|2)dz ∧ dz − zdz ∧ zdz](1 + |z|2)2

=i

2π

dz ∧ dz(1 + |z|2)2

and the conclusion follows from the transitivity of SU(n+ 1) on TCPn.

Definition 3.5.2. ω is called the Fubini study metric in CPn and is denoted ωFS . It depends on thechoice of coordinates on Cn+1.

50

Similarly for a holomorphic vector bundle E −→ X with a Hermit metric h, one has the line bundle

L π−1(E) E

P(E) X

i π

π

Π

where P(E) = (E\zero sections)/C∗ and L is denoted by L = OP(E)(−1). The composition π i is calledthe blow up of E at its zero section. Here π−1(E) is the fibre product or pullback of π and Π. LetF = (PE)x∈X be the fibre of π at x and f : F → PE the inclusion. Then f∗c1(| |2h) is a positive (1, 1)-form,where c1(| |2h) = i

2π∂∂log| |2h. Hence c1(| |2h) is a (1, 1)-form on P(E) that is positive in the vertical directionof π of X, where X is Kahler with Kahler form ωX . Hence P(E) is also Kahler.

Definition 3.5.3. OEϕ(h) := L−k where L−k := (L∨)⊗k.

Note that given a vector bundle Eϕ−→ X, then 1ϕ = ϕEϕ(1) = L∨ = L−1 is a holomorphic line bundle over

P(E).

Consider the compactification E = P(E ⊕ O) ⊇ E, Eϕ−→ X and E ⊕ O ϕ−→ X are vector bundles over X,

and E is open in P(E⊕O). We see that the blow up of E (or E) along its zero section lies in P(ϕ−1(E)) andhence it is Kahler.

51

52

Chapter 4

SHEAF COHOMOLOGY

4.1 Sheaves

Definition 4.1.1. Let X be a topological space and A an abelian category. A presheaf F on X is acollection of objects F(U) of objects in A, for each open subset U ⊆ X, and a collection of morphisms

ρUV : F(U) −→ F(V )

σ 7→ σ|V = ρUV (σ)

for each inclusion of open subsets V → U such that

ρUV = ρVW ρUV .

The last equality is known as compatibility.

A presheaf F is called a sheaf if it is saturated, i.e., if it satisfies the following condition: Let si ∈ F(Ui)be a collection of sections such that

si|Uij = sj |Uij ,

where Uij = Ui ∩ Uj , then there exists a unique section s ∈ F(∪Ui) such that s|Ui = si.

Definition 4.1.2. A morphism of (pre)sheaves is a map ϕ : F −→ G which associates to each opensubset U ⊆ X a morphism

ϕU : F(U) −→ G(U)

such that for every V ⊆ U open

ρUV ϕU = ϕV ρUV . (compatibility)

Example 4.1.1. Sheaves of sections of vector bundles (C∞, Ch, Cω for real analytic, O, etc).

53

Lemma 4.1.1. If F is a presheaf over X then there exists a unique sheaf Fsh along with a morphismF −→ Fsh that factors thorough all morphisms F −→ G to a sheaf G.

F G

Fsh

ϕ ∃!

If X is a topological space, its structure sheaf C0X associated to each open subset U is given by the space

of continuous functions on U . If X is an algebraic variety, its structure sheaf OX is given by the space ofregular functions on (Zariski) open subsets. A complex algebraic manifold is also a complex manifold, and

we write OalgX and Ohol

X to distinguish the structures:

OalgX −→ Zariski topology,

OholX −→ ordinary topology.

Definition 4.1.3. Let A be the sheaf of rings over X. A sheaf F is called an A-module if for every openset U , F(U) is a module over A(U), compatible with the restriction maps.

Remark 4.1.1. All notions from module theory carry over: Hom, Ker, Im, CoKer, direct sums, tensorproducts, exact sequences and homology groups, etc.

Definition 4.1.4. The A-module A⊕n = A is said to be free of rank n ∈ N. A sheaf that is locallyisomorphic to A⊕n is called locally free of rank n.

Remark 4.1.2. There exists a bijection between vector bundles of rank n and locally free sheaves of rankn.

If A is a sheaf of fields, the rank one sheaves (invertible sheaves with respect to A) form a group under tensorproduct, called the Picard group PicOX (X).

Definition 4.1.5. Let (f, f#) : (X,A) −→ (Y,B) be a morphism of ringed spaces, i.e., f : X −→ Y is acontinuous function, and

f#: B(V ) −→ A(f−1(V ))

is a morphism for every open subset V compatible with restrictions. The pullback sheaf f∗G of a B-moduleG is defined as follows: set f (∗)G(U) = lim−→f(U)→V G(V ) and then set f∗G be the sheaf associated to the

presheaf f (∗)G ⊗f(∗)B A.

Example 4.1.2. The sheaf F of holomorphic sections of a holomorphic vector bundles F over X,

iX : x → X

(1) i(∗)x F =: Fx (the stalk of F at x and its elements are germs of sections of F at x).

54

(2) i∗xF = Fx has finite dimension over C.

Definition 4.1.6. A sheaf of modules M on an algebraic variety (X,OX) is said to be (quasi)-coherentif it is locally isomorphic to the cokernel of a morphism of free sheaves (of finite rank).

Easy fact: f∗ preserves (locally) free sheaves, rank and invertibility. In particular, f∗ gives a homomorphismof Picard groups.

Definition 4.1.7. A short sequence of sheaves

0 −→ F −→ G −→ H −→ 0

is said to be exact if and only if it is exact on the level of stalks.

Let (f, f∗) : (X,A) −→ (Y,B) be a morphism of ringed spaces, i.e., f : X −→ Y is a continuous map, A andB are shaves of rings. For a sheaf of A-modules F on X, the direct image sheaf f∗F of F on X is the sheafof B-modules on Y given by V 7→ F(f−1(V )). Recall that for a sheaf of B-modules G on Y , its pullback isdefined as follows: Set f (∗)G(U) = lim−→f(U)⊆V G(V ), and then set (f∗G) to be the sheaf associated with the

presheaf f (∗)G ⊗f(∗)B A.

Example 4.1.3. Let F be the sheaf of holomorphic sections of a vector bundle F and i : x −→ (X,O).

(1) i(∗)F =: Fx is called the stalk of i, and it equals the set of germs of sections of F .

(2) i∗F = Fx, the fibre of F at x, is a finite dimensional vector space.

Definition 4.1.8. Recall that a sheaf of modules on an algebraic variety (X,OX) is said to be quasi-coherent (resp. coherent) if it is locally isomorphic to the cokernel of a morphism of free shaves (resp. offinite rank). By a free sheaf we mean a sheaf O⊕nX , where n is a cardinal number.

Fact 4.1.1. f∗ preserves local freeness invertibility, in particular f∗ gives a homomorphism of Picard groups,where

Pic(X) = group of invertible sheaves ∼= holomorphic line bundles

A short sequence of shaves 0 −→ F −→ G −→ H −→ 0 is said to be exact if it is exact at the level of stalks.

Remark 4.1.3. Ker, CoKer, ⊗ and f∗ preserve the property of being (quasi-)coherent. However, f∗ does not.

Example 4.1.4.

(1) f : C −→ 0. Then f∗OalgC = C[z] which is not finite dimensional.

(2) i : C∗ −→ C. Then i∗OalgC∗ = C[z, z−1] is not of finite type over C[z].

55

Construction: Let X be an affine variety over K, and M a module over K[x]. Then

X = zeroes of f1, . . . , fl over CN .

We have

K[X] := C[z1, . . . , zN ]/ 〈f1, . . . , fl〉

Then U 7→ M ⊗K[X] OX(U) is an OX -module and this correspondence preserves tensor product, exactness,

etc. Call this OX -module M . In part, M is quasi-coherent (and coherent if M is of finite type) and anyquasi-coherent OX -module is of this form.

Example 4.1.5 (Important). Ideal sheaf IY → OX corresponding to the subsheaf of OX vanishing onY → X (i.e., Y is an algebraic subvariety).

We normally assume that the ground field K is algebraically closed. Then the Nullstellensatz tells us thatV (I) := sup(OX/I) ⊆ X is nonempty if and only if I 6= OX . Hence the ringed space (V (I),OX/I) isidentified with the subscheme of X corresponding to I. A ringed space locally isomorphic to subschemesof affine spaces is called al algebraic scheme.

Definition 4.1.9 (Associated fibre spaces). Let A be a quasi-coherent sheaf of a OX -algebra of finite type(i.e., locally generated by finitely many sections as OX -algebras). We define a scheme S = SpecmXA anda morphism π : S −→ (X,OX) as follows: Let X be affine. Then set

SpecmXA := SpecmA(X) := maximal ideals in A(X).

Recall that if R = A(X) and M is a maximal ideal, then R/M = K if K is algebraically closed.

Let f : S −→ K be a regular function. Then f = 0c = a basis of open sets, form the Zariski topol-ogy. Let π be the dual to the K-algebra homomorphism K[X] −→ A(X). If D(f) = x ∈ X / f(x) 6= 0for f ∈ OX(X) then by the quasi-coherence of A, we have A(D(f)) = A(X) ⊗K[X] K[D(f)]. So thatSpecmA(D(f)) = π−1(D(f)).

Special case: Given a coherent sheaf of OX -modules F , let A = SymOXF . Then the associated fibre spaceis called the vector fibre space, denoted by π : V(F) −→ X. Here, Sym means the symmetric product⊕k≥0(SymkF). Note that

V(F)x = (Fx/MxFx)∨

Example 4.1.6 (for algebraic geometry).

(I) Definition of normal and tangent bundles (cones):

– The model for tangent vector space at a point is given by Specm(K[ε]/ε2).

– The Zariski tangent space: Let x ∈ X be a point of al algebraic variety. Then TxX :=(Mx/M2

x)∨.

– The normal bundle of Y → X (algebraic subvariety): Let I be the ideal sheaf of Y , thenI/I2 = I⊗OX OY . The normal vector bundle (or normal bundle) of Y in X is I/I2. It is denoted

by NY |X(π−→ Y ).

56

– The normal cone of Y in X is defined by

CY |X := SpecmY

(⊕k≥0Ik/Ik+1

)NY |X

Y

An easiest definition of the tangent bundle to an algebraic variety X is that it is the normal cone to

the diagonal X∆→ X×X. These are functorial objects since f : X −→ Y , f×f : X×X −→ Y ×Y ,

so Tf : TX −→ TY . And it coincides with dxf : TxX −→ Tf(x)Y , for every x ∈ X.

– We say that x ∈ X is a smooth point if Cx(X) = TxX, and X is smooth (non-singular) if allpoints are.

(II) The cotangent sheaf to X is defined as the conormal sheaf to the diagonal in X×X. Its local sectionsare local forms on TX and such a form d gives a map

Mx/M2x∼−→ Ω′X(x) := Ω′X,x/(Ω

′X,x ⊗Mx)

where Ω′X = differential on OX.

(III) Blowing up a subscheme: Let I → OX be an ideal sheaf defining a subscheme Y → X, A =⊕k>0Ik. Then σ : X = proj(A) −→ X is called the blow up of X along Y , where proj(A) =Specm(homogeneous decomposition of A). By functoriality, σ−1(Y ) is the projection of the algebraA ⊗OX OY = ⊕k≥0Ik/Ik+1, i.e., σ−1(Y ) −→ Y is the projectivization of the normal cone CY |N , i.e.,0 = ⊗OX .

Definition 4.1.10. A sheaf is torsion free is ⊗OX = 0, i.e., it is supported on a subvariety.

Fact 4.1.2.

• Any torsion free OX -module F admits a resolution, i.e., a birational morphism σ : Y −→ X such thatσ∗F is locally free.

• (Hironaka) Any variety X (any rational map X −→ Y ) admits a resolution of singularities by repeatedlyblow ups along smooth centres (i.e., smooth subvariety)

X X

Y

σ

57

4.2 Cohomology of sheaves

Let X be a topological space. We consider sheaves F i together with morphisms di : F i −→ F i+1 such thatdi+1 di = 0 for every i. Such set of sheaves and morphisms (Fi, di) is called a complex of sheaves over X.It is also called a resolution of a sheaf F if there exists an inclusion ι : F −→ F0 such that i(F) = Ker(d0)and Ker(di+1) = Im(di), for every i.

(1) Cech resolution: Let F be a sheaf over X, Uii∈N a covering of X. For every finite subset I ⊆ Nset UI = ∩i∈IUi, jI : UI → X and

FI = (jI)∗(F|UI ) (extension by zero outside UI)

Define Fk := ⊕|I|=k+iFI and d : Fk −→ Fk+1 by

(dσ)j0...jk+1=∑

i

(−1)iσj0 . . . ji . . . jk+1|U∩UI

where j0 ≤ j1 ≤ · · · ≤ jk+1, σ = (σI), σI ∈ FI(U) and |I| = k + 1. Lastly, we define ι : F −→ F0 by

ι(σ)i = σ|U∩Ui for σ ∈ F(U).

Proposition 4.2.1. This is a resolution.

(2) de Rham resolution: Let Ak be the sheaf of C∞ (R or C-valued) differential forms of degree k (ona real or complex manifold). The d-Poincare Lemma says that the complex (Ak, d) is a resolution ofKer(d0) = R (resp. C), constant sheaves over X.

(3) Dolbeault resolution: Let E be a holomorphic vector bundle over a complex manifold X and E itssheaf of holomorphic sections (i.e., E = OX(E)). Let A0,q be the sheaf of C∞-sections of Ω0,q

X ⊗ E.Generalizing the d-Poincare Lemma, we get that (A0,q(E), ∂) is a resolution of Ker(∂0) = E = OX(E)(a coherent OX -module).

58

4.3 Coherent sheaves

Let OX be the space of functions on X. The “sheaf version”of this space means that to every open subsetU ⊆ X it is associated an space of forms on U .

Example 4.3.1. To an algebraic variety X we associate the space of rational 1, 1-forms on X without poleson U . To any complex manifold X we associate the space of holomorphic functions on U .

A coherent sheaf is free if it is of the form O⊕nX . An ideal sheaf I is just a subsheaf of OX which is an idealof OX(U) (a ring) for every U . Normally assume I 6= OX . We have a short exact sequence

0 −→ I −→ OX −→ OZ(I) −→ 0,

where Z(I) is a scheme equal to Z(I) = spec(OX/I) ( X, which is nonempty. Note that OZ(I) and I areexamples of coherent sheaves over X, and OZ(I) is called the torsion part. Any coherent sheaf is locally afinite direct sum of these factors. If there are no factors of the form OZ(I), i.e., not supported on a proper

subvariety, then it is called torsion free. Any torsion free sheaf admits a resolution f : Xres−→ X such

that f∗ of the sheaf is locally free (i.e., a vector bundle). This is because given an ideal sheaf I defining asubscheme Y ⊆ X, A = ⊕k>0Ik is an algebra over OX . Then σ : X = proj(A) −→ X (an isomorphismoutside Y ), called the blowup of X along Y , is a birational map to X that replaces Y by a subscheme ofcodimension 1, i.e., σ−1(Y ) is locally given by one equation and σ−1(Y ) −→ Y is the projectivization of thenormal cone CY |X .

Given a collection of sheaves F i over X with morphism di : F i −→ F i+1 such that di+1 di = 0 for everyi. It is a resolution of a sheaf F if there exists an inclusion i : F −→ F0 such that j(F) = Ker(d0) andKer(di+1) = Im(di) for every i.

(1) There exists a sheaf X, Uii∈N covering of X. For every finite set I ⊆ N, set UI = ∩i∈IUi, jI : UI → X,and FI = (jI)∗(FUI ) extended by zero outside UI . Set Fk = ⊕|I|=k+1FI and d : Fk −→ Fk+1 by

(dσ)j0···jk+1=∑

i

(−1)i(σj0···ji···jk+1)|U∩UI ,

where σ = (σI), σI ∈ FkI (U), and j : F −→ F0 is given by j(σ)i = σ|Ui∩U for σ ∈ F(U).

Proposition 4.3.1. This is a resolution, where Fi|Ui has trivial cohomology.

(2) de Rham resolution: Let Ak be the sheaf of C∞ (R or C)-valued k-forms. The Poincare Lemmasays that the complex (Ak, d) is a resolution of Ker(d0) = R or C, constant sheaves.

(3) Let E be a holomorphic vector bundle and E its sheaf of holomorphic sections. Let A0,q(E) be thesheaf of C∞-sections Ω0,q

X ⊗ E. The ∂-Poincare Lemma implies that (A0,q(E), ∂) is a resolution ofKer(∂0) = E .

59

4.4 Derived functors

Given a functor F : C −→ C′ between abelian categories such that C has enough injectives.

Definition 4.4.1. For every object M ∈ C, there exists an object RiF (M) for every i ≥ 0 in C′, unique upto isomorphisms, satisfying the following conditions:

(1) R0F (M) = F (M),

(2) Every short exact sequence 0 −→ A −→ B −→ C −→ 0 gives rise to a long exact sequence in C′

0 −→ F (A) −→ F (B) −→ F (C) −→ R1F (A) −→ · · ·· · · −→ RiF (A) −→ RiF (B) −→ RiF (C) −→ Ri+1F (A) −→ · · ·

Remark 4.4.1.

(1) LetM0, i : A −→M0 be an acyclic resolution of A (i.e., Rj+1F (Mk) = 0 for every j, k ≥ 0). Then

RjF (A) = Hj(F (M0)) := CoKer(di−1F(Mi−1) −→ F(Mi))

(2) For a sheaf F over X there exists an acyclic resolution (called the Godement resolution for F).

Definition 4.4.2. A fine sheaf F os a sheaf of A-modules where the sheaf of algebra A admits a partitionof unity: for every open covering Ui, there exists fi ∈ F(Ui) such that

∑i fi = 1 (this sum is locally finite).

Proposition 4.4.1. Hi(X,F) = 0 for every i ≥ 0 for such F .

Corollary 4.4.1.

(1) Let X be a real (complex) manifold, then

H∗(X;R) =Ker(d)

Im(d)= H∗dR(X;R),

where the same equality holds if we replace R by C.

(2) Given a holomorphic vector bundle E −→ X and E its sheaf of holomorphic sections. Then

H∗(X, E) =Ker(∂ : A0,q(E) −→ A0,q+1(E))

Im(∂ : A0,q−1(E) −→ A0,q(E))

Corollary 4.4.2 (Grothendieck Vanishing Theorem). Given a holomorphic vector bundle E −→ X, wehave Hq(X,E) = 0 for q > n = dimCX.

Consider a Cech resolution

Ker(d) = F −→ F0 d−→ F1 −→ · · · −→ Fk −→ · · ·

where the map F −→ F0 is given by σ 7→ σ|U∩Ui and Fk = ⊕|I|=k+1(jI)∗(F|UI ).

60

Theorem 4.4.1. If Hq>0(UI ,F) = 0 for every I ⊆ N then

Hq(X,F) = Hq(U,F) := Hq(Γ(F) = F(X), dX),

e.g., if X is a finite dimensional manifold C∞, then there exists a good cover (UI contractible for every I)and so

Hq(U,Z) = Hqcell(X,Z)

where U is in an open cover and Hqcell(X,Z) denotes the cellular cohomology, which is computing via nerves

of the open covering.

61

62

Chapter 5

HARMONIC FORMS

5.1 Harmonic forms on compact manifolds

Let (X, g) be a Riemannian manifold, where X compact is a blanket assumption. Then we have a metric( , ) on ∆∗T∨X,x. We assume X is oriented. Let α, β ∈ Ak : C∞(AkT∨X). Then

〈α, β〉 =

∫

X

〈α, β〉x dVol(x)

gives an L2-metric on Ak. We also have a pointwise isomorphism p : ∆n−kT∨x∼−→ Hom(∆kT∨x ,∆

nT∨x ) givenby v 7→ v ∧ −, where ∆nT∨x = RdVol(x), and an isomorphism m : ∆kT∨x

∼−→ Hom(∆kT∨,R) given bye 7→ 〈e, 〉∆kT∨ .

Definition 5.1.1. The Hodge Star Operator is given by

∗ = p−1 m : ∆kT∨x∼−→ ∆n−kT∨x

and the associated global isomorphism by

∗ : ∆kT∨∼−→ ∆n−kT∨

Ωk(X) −→ Ωn−k(X)

We extend ∗ to complex-valued forms by extending 〈 , 〉 to Hermit metrics on ∆kC(T∨ ⊗ C) = (∆k

RT∨)⊗ C.

We get (α, β)xdVol(x) = αx ∧ βx and so

〈α, β〉 =

∫

X

α ∧ ∗β

is the L2-metric on AkC = Ak ⊗ C. In the case X is complex, AkC = ⊕p+q=k ⊕Ap,q, Ap,q = C∞(Ωp,qX ).

Fact 5.1.1. The Stokes Theorem implies that (α, d∗β) = (dα, β) where d∗ := (−1)k ∗−1 d∗.

63

In part, if n is even then d∗ = − ∗ d∗. Similarly,

Fact 5.1.2. ∂∗ = −∗ ∂∗ and ∂∗

= −∗ ∂∗ are formal adjoint of ∂ and ∂ with respect to the L2 metric in AkC.

Proof: (∂α, β) =∫X∂α ∧ ∗β = −

∫X

(−1)|α|α ∧ ∂ ∗ β = −∫X

(−1)|α|α ∧ ∗ ∗−1 ∂ ∗ β = (α, ∂∗β).

More generally, if (E, h) is a Hermitian vector bundle then there exists a C-anti linear isomorphism of vectorbundles given by h : Ω0,q

X ⊗E = Ω0,q(E) −→ (Ω0,q ⊗E)∨ ∼= Ωn,n−q ⊗E∨, where ∆2nX = Ωn,n = RdVol(x). So

it gives am antilinear isomorphism

∗E : Ω0,q(E)∼−→ Ωn,n−q(E∨) = KX ⊗ Ω0,n−q(E∨)

called the Hodge Star, where KX = Ωn,0 = ∆nCT∨X (holomorphic line bundle) is called the canonical

bundle of a complex manifold X.

Fact 5.1.3. ∂∗X = (−1)q ∗−1

E ∂KX⊗E∨ : A0,q(E) −→ A0,q−1(E) is the formal adjoint of ∂E .

Fact 5.1.4. (d∗)2 = (∂∗E)2 = (∂∗)2 = 0.

Definition 5.1.2. Let (X, g) be a Riemannian manifold,

∆ := dd∗ + d∗d = (d+ d∗)2

Definition 5.1.3. Let (X,h) be a Hermitian manifold,

∆∂ := ∂∂∗ + ∂∗∂ = (∂ + ∂∗)2,

∆∂ := ∂∂∗

+ ∂∗∂ = (∂ + ∂

∗)2.

If further E −→ X is a holomorphic vector bundle with a Hermit metric, we write ∆E for ∆∂E= (∂E +∂

∗E)2.

From construction,

〈α,∆dα〉 = ||dα||2 + ||d∗α||2

and analogously for the Hermitian case.

Corollary 5.1.1. Ker(∆d) = Ker(d) ∩Ker(d∗).

Definition 5.1.4. An element of Ker(∆d) is called harmonic, i.e., it is killed by d and d∗.

64

Theorem 5.1.1 (Main Theorem of the Course). Let

(F, φ) =

(⊕kΩkX ,∆d

)for the Riemannian case,(

⊕qΩ0,q(E),∆∂E

)for the Hermitian case.

where φ : F −→ F is an automorphism, and F = ⊕kΩkX or F = ⊕qΩ0,q(E).

65

5.2 Some applications of the Main Theorem

Theorem 5.2.1 (Riemannian case). Let Hk be the space of harmonic k-forms. Then the mapHk −→ Hk

dR(X,R (or C)) given by α 7→ [α], which is well defined since dα = 0, is an isomorphism.

Proof: By the Main Theorem, β ∈ Ak can be written as α + ∆γ = α + dd∗γ + d∗dγ, where α +dd∗α is d-closed. Since β is closed we have d∗dγ is closed and hence it belongs to (Im(d∗))⊥. So0 = (dγ, dd∗dγ) = ||d∗dγ||2. Similarly, we get the analogous theorem for ∆∂E

where we can identify

Hq(X,O(E) = E) with H0,q

∂(X,E) via the Dolbeaut isomorphism.

Theorem 5.2.2. Given a holomorphic vector bundle E −→ X, let H0,q(E) be the space of (∂E)-holomorphicforms of type (0, q) with values in E. Then the map H0,q(E) −→ Hq(X, E) given by α 7→ [α] is an isomor-phism.

Corollary 5.2.1. If X be a compact manifold then Hq(X,R) is finite dimensional. If X is a compactcomplex manifold and E −→ X is a holomorphic vector bundle then Hq(X,E) is finite dimensional.

66

5.3 Review

Theorem 5.3.1. Let X be a compact manifold:

(F, P ) =

(⊕k ΩkX ,∆d

)if X is Riemannian,(⊕

q Ω0,q(E),∆∂E

)E −→ X is a holomorphic vector bundle

where P is an elliptic or Laplacian-like operator. This implies that

C∞(F ) = Ker(P )⊥©P (C∞(F ))

where Ker(P ) is finite dimensional.

Corollary 5.3.1.

(1) There is an isomorphism Hk ∼−→ HkdR(X,Ror C) given by α 7→ [α], where Hk is the finite dimensional

space of harmonic forms.

(2) There is an isomorphism H0,q ∼−→ Hq(X,E), where H0,q is the finite dimensional space of (0, q)-forms.

Both isomorphisms depend on the chosen metric.

Note that

HkdR(X,R) Hk(X,R)

H(H,R)

∼=1

∼=2 ∼=3

where ∼=1 is given by the de Rham isomorphism, ∼=2 and ∼=3 are given by the Poincare Lemma. Recall theDolbeaut isomorphism:

Hp,q

∂(X,E) ∼= H0,q

∂(X,ΩpE) ∼=(∗) Hp,q(X,ΩpE) =: Hp,q(X,E) (finite dimensional),

ΩpE ∼= O(∆pT ∗X ⊗ E) (space of holomorphic p-forms with values in E).

where (∗) comes from the isomorphism ΩpE = Ker∂0(Ap,0)(E) −→ Ap,1(E)). Also,

Hp,q(X) = Hp,q(X,C) = Hq(X,ΩpX) (sheaf of holomorphic functions).

It is known that ∆ = ∂∆∂ in every Kahler manifold. Then the following theorem follows:

Theorem 5.3.2. Let X be a Kahler manifold. Then

Hk(X,C) =⊕

p+q=k

Hp,q(X).

In other words, if [α] ∈ Hk(X,C) with α harmonic, then α =∑p+q=k α

p,q, where αp,q is the (harmonic)component of type (p, q).

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5.4 Heat equation approach

Given an initial distribution of heat f(x) = F (x, 0) (t = 0) on a Riemannian manifold (X, g), then the heatF (x, t) at time t is governed by (∂t + ∆X)F = 0.

Example 5.4.1. F is easily obtained for every t for S1, and in general for the torus as follows:

F (0, t) =∑

an(t)einθ.

We have

∂t + ∆θ = 0 =⇒ 0 =∑(

a′n(t) + n2an(t))einθ

=⇒ an(t) = ane−n2t, where an = an(0),

=⇒ F (θ, t) =∑

n≥0

e−n2tane

inθ.

It follows F (θ, t) −→ a0 =∫S1 f(θ)dθ = AvS1(f), where the integral is the initial distribution.

Example 5.4.2. Let X = R. Doing the same exercise as above but using Fourier transforms, we get

F (x, t) =1√4πt

∫

Re−

(x−y)24t f(y)dy =

∫

ReR(x, y, t)f(y)dy.

The function eR(x, y, t) = 1√4πt

e−(x−y)2

4t is called the heat kernel. Similarly, eRn(x, y, t) = 1√4πt

e−||x−y||2

4t .

On S1, we have e(x, y, t) =∑n e−n2tein(x−y) =

∑n e−n2teinxeiny, where e−n

2t are the eigenvalues of ∆, and

einx and einy are the eigenfunctions. In general, the existence of eX(x, y, t) is difficult to obtain analyticallybut trivial on physical grounds.

Remark 5.4.1. F (x, t) is smooth for every t > 0, i.e., immediate smoothing by heat flow.

In general, given a form α on (X, g), wish to solve

(∗)

(∂t + ∆)A(t) = 0A(0) = α

where α(t) is a form on X parametrized by t. Uniqueness of A(t) follows from:

Lemma 5.4.1. ||A(t)|| is decreasing (non-strict) for a solution of (∗).

Proof: ∂t||A(t)||2 = 2 〈∂tA,A〉 = −2 〈∆A,A〉 = −2⟨||dA||2 + ||d∗A||2

⟩≤ 0.

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Theorem 5.4.1. Let (X, g) be a compact Riemannian manifold. Then there exists Kp(x, y, t) ∈ Apx(X),depending only on (X, g) and p, called the heat kernel of X on p-forms such that

A(t) =

∫

X

Kp(·, y, t)α(y)dVol(y)

solves (∗), for every α ∈ Ap(X).

Let Tt(x) =∫XK(·, y, t)α(y)dy.

Theorem 5.4.2. Tt satisfies:

(1) Tt1+t2 = Tt1Tt2 .

(2) Tt is formally self-adjoint.

(3) Ttα tends to a C∞ harmonic form H(α) as t −→∞.

(4) G(α) =∫∞

0(Ttα−Hα)dt is well defined and yields the Green operator G, i.e.

G(α) ⊥ (harmonic forms) and α = H(α) + ∆G(α).

Proof:

(1) Holds because A(t1 + t) solves the heat equation with initial condition A(t1).

(2) ∂t 〈Ttη, Tτ ε〉 = 〈∂tTtη, Tτ ε〉 = −〈∆Ttη, Tτ ε〉 = −〈Ttη,∆Tτ ε〉 = 〈Ttη, ∂tTτ ε〉 = ∂t 〈Ttη, Tτ ε〉. Thisimplies that 〈 , 〉 is a function of t+ τ , so denote 〈 , 〉 by g(t+ τ). Therefore,

〈Ttη, ε〉 = g(t+ 0) = g(0 + t) = 〈η, Tτ ε〉 .

(3) (1) + (2) =⇒ ∀h > 0,

||Tt+2hα− Ttα||2 = ||Tt+2hα||2 + ||Ttα||2 − 2 〈Tt+2hα, Ttα〉= ||Tt+2hα||2 − ||Ttα||2 − 2(||Tt+hα||2 − ||Tt+2hα||||Ttα||)

and ||Tαα||2 converges, and therefore it is decreasing. Hence ||Tt+2hα − Ttα||2 −→ 0. It follows

Ttα −→ H(α), for some H(α) ∈ Ap(X)L2

, called the harmonic projection. Fix τ > 0, thenTtα = TτTt−τα −→ H(α) := TτH(α) as t −→ ∞. Hence H(α) is C∞ since Tτ is given by a C∞

kernel. Hence H = limt→∞ Tt is also formally self-adjoint.

(4) ||Ttα − Hα|| can be shown to decay rapidly enough so that G(α) =∫∞

0(Ttα − H(α))dt is well

defined. We verify that G is formally the Green operator:

∆G(α) =

∫ ∞

0

∆Ttαdt =

∫ ∞

0

−∂tTtαdt = α−H(α),

and for β harmonic we have

〈Gα, β〉 =

∫ ∞

0

〈(Tt −H)α, β〉 dt =

∫ ∞

0

〈α, (Tt −H)β〉 dt = 0, since (Tt −H)β = 0.

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Corollary 5.4.1. There exists an orthogonal direct sum decomposition

Ap(X) = Hp(X)⊕

d(Ap−1(X)) = Hp(X)⊕

d∗(Ap−1(X))

where Im(∆) = Im(d) + Im(d∗).

Corollary 5.4.2. There is an isomorphism Hp(X) ∼= HpdR(X) given by α 7→ [α].

Then O ∈ Ap(X) =⇒ O = α+ dd∗γ + d∗dγ, where α ∈ H(X). Also,

||d∗dγ||2 = 〈d∗dγ, α〉 = 〈dγ, dα〉 , where dα = 0.

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5.5 Index Theorem (Heat Equation approach)

Let ∆p : Ap(X) −→ Ap(X), λ ∈ R≥0. Let Epλ denote the λ-eigenspace for ∆p (finite dimensonal). The

square root√

∆ = δ is called the Dirac operator.

Lemma 5.5.1. The sequence

0 −→ E0λ

d−→ E1λ −→ · · ·

d−→ Enλ −→ 0

is exact for λ > 0.

Proof: ω ∈ Epλ =⇒ ∆p+1dω = d∆pω = λdω =⇒ dω ∈ Ep+1λ .

Now ω ∈ Epλ and dω = 0 =⇒ λω = ∆pω = d∗d+ dd∗ω =⇒ ω = d(

1λd∗ω). Then ∆d∗ω = d∗∆ω = λd∗ω.

Corollary 5.5.1.∑p(−1)pdim(Epλ) = 0.

Corollary 5.5.2. Let λpi be the spectrum of ∆p, with terms repeated n times if multi = n. Then

∑

p

(−1)ptre−t∆p

=∑

p

(−1)pe−tλ(p)i =

∑

p

(−1)p′∑

i

e−λ(p)i (t)

where∑′i is over i where λ

(p)i = 0.

Note that∑′i e−λ(p)

i (t) = dim(Ker(∆p)). Hence

X (X) =∑

p

(−1)pdim(Ker(∆p)) =∑

p

(−1)ptre−t∆(p), where e−t∆(p) = Tt,

=∑

p

(−1)p∑

i

∫

X

e(p)(x, x, t)dVol(x).

Proposition 5.5.1. e(x, x, t) ∼ (4πt)−n/2∑∞k=0 uk(x, t)tk, where uk(x, t) is explicitly given in terms of

components of curvatures.

Hence, as t −→ 0, we have

X ∼ 1

4πt

n/2 ∞∑

k=0

(∫

X

∞∑

p=0

(−1)ptrupk(x, x)dVol(x)

)tk.

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This implies

4πn/2∫

X

∞∑

p=0

(−1)trupk(x, x)dVol(x) =

0 if k 6= n/2,X (X) k = n/2.

Theorem 5.5.1 (Gauss-Bonet). Let n = dim(X) be even. Then

X (X) =

∫

X

ω,

where ω is given in a local frame by

ω = cn∑

σ,τ

(signσ)(signτ)Rσ(1)σ(2)τ(1)τ(2) · · ·Rσ(n−1)σ(n)τ(n−1)τ(n)

and cn = (−1)n/2

(8π)n/2(n2 )!.

For n = 2, ω = 18π (R1212−R1221−R2112−R2121)dA = − 1

2πRdA1212 = K

2πdA where K is the Gaussian curvature.

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BIBLIOGRAPHY

[1] Griffiths Phillip; Harris Joseph. Principles of Algebraic Geometry. Pure & Applied Mathematics. JohnWiley and Sons, Inc. New York (1978).

[2] Voisin, Claire. Hodge Theory I.

[3] Voisin, Claire. Hodge Theory II.

[4] Arapura, Danu. Algebraic Geometry over the Complex Numbers.

[5] Rosemberg, Steven. The Laplacian on a Riemannian Manifold.

[6] Yu, Yan-Lin. The Index Theorem and the Heat Equation Method.

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