COMPLEX GEOMETRY Course Notes

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 errors are responsibility of the author.

On the cover: a picture of the Riemann Sphere

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

i 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

i 2.10 Product structures on i HdR(Z) ...... 27 2.11 Questions about (compact)L 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

iii 3 COMPLEX MANIFOLDS 43

3.1 Complex manifolds and forms ...... 43 3.2 K¨ahlermanifolds ...... 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

iv 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 , i.e., − −→ U df f(z) f(z0) f (z ) = (z ) = lim − 0 0 dz 0 z z0 z z0 → − exists for every z0 . We shall denote this by f (U). If S C is any subset, we shall say that f is holomorphic on S (∈f U (S)) if f is holomorphic on∈ a O open neighbourhood⊆ of S. ∈ O

∂f ∂f 2 If the function f is R-diﬀerentiable on U then ∂x dx + ∂y makes sense and df(x, y) HomR(Tz=x+iyU, R ). Recall that ∈ dz = dx + idy and dz = dx idy − Using these expressions, we can write the diﬀerential df as

df = 1 ∂f i ∂f dz + 1 ∂f + i ∂f dz = ∂f dz + ∂f dz 2 ∂x − ∂y 2 ∂x ∂y ∂z ∂z Notice the following relations

∂f ∂f ∂f ∂f ∂z = ∂z and ∂z = ∂z Recall that

f is complex diﬀerentiable

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

∂u =⇐⇒i ∂v ∂z − ∂z

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

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

Theorem 1.1.1 (Cauchy). f (U) if and only if f = 0, for every V U simple connected. ∈ O ∂V ⊂⊂ R

Theorem 1.1.2 (Cauchy’s Integral Formula). z V U if and only if 0 ∈ ⊂⊂ 1 f(z) f(z ) = dz. 0 2πi z z Z∂V − 0

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

1 2π f(z ) = Avg (f) := f(z + eiθ)dθ. 0 ∂V 2π 0 Z0

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

Proof: Let V = B(z0), z0 C. We show that f 0(z0) = 0. Using the Cauchy’s Integral formula, we have ∈ 1 f(z) f 0(z ) = 0 2πi (z z )2 Z∂V − 0 Notice that f is bounded on ∂V . Then f(z) M on ∂V for some M > 0. So we have | | | | ≤ 1 f(z) 1 f(z) f 0(z ) = dz | | dz | 0 | 2π (z z )2 ≤ 2π2 z z 2 Z∂V − 0 Z∂V | − 0| 1 M = f(z) dz dz 2π 2 | | ≤ 2π2 Z∂V Z∂V M M = 2π = 2π2 ·

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

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

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

2 Theorem 1.1.3 (Local Structure of f (U)). If f (U) is non-constant at z U. Let ∈ O ∈ O 0 ∈ m = min n > 0 / f (n)(z ) = 0 . { 0 6 }

Then there exists a bi-holomorphic function ϕ : V W from a neighbourhood of V of z0 to a neighbourhood W of 0 with ϕ(z ) = 0 such that −→ ∈ C 0 f(z) f(z ) = ϕ(z)m, for every z V. − 0 ∈

m Proof: Note that f(z) f(z0) = (z z0) g(z) with g(z0) = 0 and g (U). Since the quotient f(z) f(z0) − − 6 ∈ O − is bounded on U z and continuous at z , we have by the Riemann Extension Theorem that z z0 0 0 − − m 1 f(z) f(z0) (z z ) g(z) = − is holomorphic on U. Proceeding this way, we have that g(z) (U). 0 − z z0 − − ∈ O We study several cases: If n = 1 then f 0(z ) = 0 and by the Inverse Function Theorem we can choose 0 6 ϕ(z) = f(z) f(z0). Now assume n = 1. Since g(z0) = 0 then g(z) = 0 on a neighbourhood of z0. So we can write−g = hm on a neighbourhood6 V . We have 6 6

f(z) f(z ) = [h(z)(z z )]m − 0 − 0 m with ϕ0(z ) = h(z ) = 0. Hence, up to a local change of coordinates, f is locally of the form z z for 0 0 6 7→ some m. Such a number m is called the ramiﬁcation degree of f at z0.

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

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

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

3 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 a −→ ∈ neighbourhood V of z0 such that

α β f(z) = ∞ a (x x ) (y y ) for every z V. α,β=0 α,β − 0 − 0 ∈

Similarly, f is said to be complexP analytic on U if for every z0 U there exists a neighbourhood V of z0 such that ∈ n f(z) = ∞ a (z z ) for every z V. n=0 n − 0 ∈ In both cases the equality means normalP convergence in U, i.e., uniform convergence on compacts in U.

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

Proof: We know that 1 f(w) f(z) = dw. 2πi w z Z∂Dr (z0) − On the other hand,

n 1 1 1 1 ∞ z z = = 0 . z z0 − w z w z 1 − w z0 w z0 w w0 ! n=0 − − − − − X − It follows that 1 ∞ (z z )n ∞ f(z) = f(z) − 0 dw = a (z z )n 2πi (w z )n+1 n − 0 ∂Dr (z0) n=0 0 n=0 Z X − X where 1 f(w) 1 a = dw = f (n)(z ) n 2πi (w z )n+1 n! 0 Z∂Dr (z0) − 0 and w z = r on ∂D (z ). | − 0| r 0

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

1 1 Proof: Suppose f − (0) is not discrete. Let γ be an isolated point in f − (0) and consider the Taylor expansion of f about γ, ∞ f (n) f(z) = (z γ)n n! − n=0 X 1 for every z Dr(γ), where r is the radius of convergence of the series. Since γ is not isolated in f − (0), ∈ 1 there exists z f − (0) D (γ). We have ∈ ∩ r ∞ f (n) 0 = (z γ)n n! − n=0 X

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

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

Lemma 1.2.1 (Schwartz). If f (D) and f M on ∂D, f(z) M z on ∂Dr for every r ( , 1), then f(z) = Nz; where N < M. ∈ O ≤ | | ≤ | | ∈ − | | Another version: If f(z) M on D and f(0) = 0, then f(z) = Nz with N < M. Here, D | | ≤ | | denotes the Poncar´edisk, i.e., the disk z C : z 2 < 1 endowed with the metric { ∈ | | } z w 2 δ(z, w) = 2 | − | . (1 z 2)(1 w 2) − | | − | |

5 1.3 Complex Analysis in several variables

Let U Cn be an open subset. ⊆

Definition 1.3.1. A function f : U C in C0 (U)(R-differentiable on U) is called holomorphic, denoted −→ R f (U), if for every u U, the differential dfu Hom(TuU, C) is a C-linear map. ∈ O ∈ ∈ We prove as before the following result:

Theorem 1.3.1. The following conditions are equivalent: (1) f (U). ∈ O (2) For every z U, f has the form 0 ∈ I f(z0 + δ) = aI δ (normal convergence) XI where I = (i , . . . , i ) and zI = zi1 zin . 1 n 1 ··· n (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 n dζ dζ f(ζ) 1 n 2πi ζi ai =αi ζ1 z1 ∧ · · · ∧ ζn zn Z| − | − − where δD = ζ a = α ∂D. {| i − i|} i ⊂

Note that if f 0 C0 (U) then ∈ R

∂f ∂f 1 ∂f ∂f 1 ∂f ∂f df = dxi + dyi = i dzi + + i dzi ∂xi ∂yi 2 ∂xi − ∂yi 2 ∂xi ∂yi P P P P n+1 n+1 Theorem 1.3.2. Let f C such that f(0) = 0. Write C = (w, z1, . . . , zn) = (w, z) . If f 0 on the w axis (z = 0) then∈ on some neighbourhood V of 0, we have { } 6≡ − d d 1 f = (w, z)(w + a (z)w − + + a (z)) 1 ··· d where g is never zero on V .

d d 1 Denote p(w, z) = w + a (z)w − + + a (z). Hence locally we have zero(f) = zero(p). It follows that 1 ··· d the 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 ´etalecover that covers the hyper- plane w = 0 = (z , . . . , z ) . So by induction we see that: { } { } { 1 n }

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.

6 Theorem 1.3.3 (Riemann Extension Theorem). Part I: If f is a holomorphic function and bounded outside an analytic subset of codimension 2 or • higher, 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 = ∆(r0), such that r0 < r and V U, then every f {(U V ) extends| | to a holomorphic| | } function on U. ⊂⊂ ∈ O − (2) If S U Cn has complex codimension greater or equal than 2, where S is an analytic subset and f⊆ ⊆(U S), then f extends to a holomorphic function on U. ∈ O −

Proof: We only proof the second part.

(1) Take a slice z = const. Then U V = r0 < z < r on this slice. Set 1 − { | 2| }

1 f(z1, w2) F (z1, z2) = dw2 2πi w2 =r w2 z2 Z| | −

Hence F : U C is holomorphic in z1 since −→ ∂f ∂F = 0 = ∂z1 ⇒ ∂z1

and clearly also in z2 (Cauchy’s Integral Formula). Moreover, F = f on U V by the Cauchy’s Integral Formula. − (2) The a 2-dimensional slice and apply (1).

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

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 (U), where U is connected (or arcwise connected), then f 0 on U. ∈ O ≡

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

7 8 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 open subsets 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 a complex 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:

n n+1 (1) CP = [z0 : : zn] = C 0 / , where { ··· } − { } ∼

(z0, . . . , zn) (z0 , . . . , z0 ) z = tz0, for some t C∗. ∼ 0 n ⇐⇒ ∈ n Let U0 = [z0 : : zn]/ z0 = 0 . Let ϕ0 : U0 C be the map given by { ··· 6 } −→ z z [z : : z ] 1 ,..., n 0 ··· n 7→ z z 0 0 n which we shall call the 0-th aﬃne chart. Note that there exist n + 1 aﬃne charts that cover CP .

n n (2) The compact complex torus C /Γ, where Γ ∼= Z . (3)

1 PGL(1, 0) = Aut(CP ) = set of Moebius transformations az + bz = 0 1 / ad bc = 0 / 1 cz + dz − 6 {± } 0 1

The following sets are Riemann surfaces:

1 (4) CP .

9 (5) C and C∗ = C 0 . − { } (6) The half-plane model H = z / Im(z) > 0 . { } (7) The Poincar´edisk model ∆ = D = z / z < 1 and ∆∗ = ∆ 0 . The models H and D are related by the map { | | } − { } z a z − 7→ z 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 an 1 elliptic curve. The manifold CP is called a rational curve.

1 Note that CP has positive curvature, C has zero curvature (in other words, C is said to be ﬂat), and H and D have negative curvature. Note that Γ is a lattice n + α / α H . The parameter space of elliptic { ∈ } curves 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 a submanifold (only need to check df(p) = 0 for every p C to have C smooth). 6 ∈ 2 (2) xd + yd + zd gives the Fermat curve of degree d in CP . It is smooth since df = (0, 0, 0) on C, d 1 n 1 n 1 6 df = (x − dx, y − dy, z − dz).

10 2.2 Holomorphic maps

1 Definition 2.2.1. Let f : X Y be a continuous map of topological spaces, the inverse image f − (p) is called 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 they coincide on a non-discrete subset of S , then f f−→. 1 1 ≡ 2

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 is −→ 1 compact. Then f is surjective and S2 is compact. Furthermore, if f is proper (i.e., f − (C) is compact for every compact set C) and discrete with finite fibres (i.e., a finite map) then f is called a branched covering map (at a branch f looks like z zd, where d is called the branched degree). 7→

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 atPp is

r (f) = mult (f) 1. p p − A map is said to be unramiﬁed if r (f) = 0 for every p S. p ∈

11 2.3 Meromorphic functions and diﬀerentials

Deﬁnition 2.3.1. A meromorphic function on a Riemann surface Z is a holomorphic function on an open 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

∞ i f(z) = aiz X−∞ with ai = 0 for every i < n Z. ∈ (c) f = g/h, where g, h (p), g(p) = 0 and h(p) = 0. ∈ O 6

The set of such functions is denoted (Z). We have M hol 1 f (Z) f : Z CP ∈ M ⇐⇒ −→ and that 1 poles of f = f − ( ) ∞ Example 2.3.1.

(1) A non-constant polynomial defines a meromorphic function from 1 with pole order at equal to deg(f) 1. CP ∞ ≥ (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. (Z) is a ﬁeld. M

A finite map of Riemann surfaces f : Z Z corresponds to a finite field extension 1 −→ 2

f ∗ : (Z ) , (Z ). M 2 → M 1 Definition 2.3.2. A meromorphic differential on a Riemann surface Z is a holomorphic differential ω on an open U Z whose complement Z U is discrete and consist of poles of ω. Locally, ω = fdz even at a 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 ⊂⊂

ω = Resp(ω) ∂V p V Z X∈

12 We shall denote the space of meromorphic diﬀerentials by 0(Z). M

Theorem 2.3.2. ω 0(Z), p Z Resp(w) = 0. ∈ M ∈ P Corollary 2.3.1 (Sum rule or product rule, Reciprocity Theorem). If Z is compact and f (Z), then ∈ M #zero(f) = #poles(f) on Z counting multiplicity.

1 1 Corollary 2.3.2. If f (CP ) then f is rational. Hence (CP ) = C(Z). ∈ M M

Corollary 2.3.3. 0(CP0) = C(Z)dz. M

Let S be a Riemann surface and U S a relative compact open subset of S with good ∂U. Let ω be a ⊂⊂ meromorphic diﬀerential (ω 0(S)). Then we have ∈ M

ω = Respω ∂U p U Z X∈ If ∂U = (so U = S is compact) then ∅ Respω = 0 p S X∈ We have that if S is compact then = R(S), the set of rational functions (later, we are going to study this in detail). S

1 1 1 Example 2.3.2. (P1) = R(P1), where P1 = CP . Note that CP is compact and CP = C C, where there is a map betweenM charts ∪ 1 w= z z C w C ∈ 7→ ∈ df Since p S Respω = 0, we have ω = f , and Respω = ordpf. ∈ P Recall that f (P1) if and only if f : P1 P1 is holomorphic. ∈ M −→

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

hol P2 p − { }

hol C p − { }

13 1 2 2 2 1 This shows that C ∼=hol P . For example, we have the Fermat curve z0 + z1 = z2 . If P = [u : v] , then deﬁne a map z = (u v)2, z = 2uv and z = (u + v)2. { } 0 − 1 2

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

1 Example 2.3.4. S1 = C∗, S2 = D D. These two examples are not biholomorphic Riemann surfaces. − 2 Neither D∗ is biholomorphic to C∗. If so, then a biholomorphic function D∗ C∗ produces an extension −→ D S, getting a contradiction. −→

The Riemann surfaces H and D are biholomorphic via the map w a ω − 7→ w a −

14 2.4 Weierstrass P -function on C

Consider the torus C/Γ, where Γ = 1, τ τ = Z Zτ h i ∈C ⊕ is a lattice in C. Deﬁne the Weirstrass P -function by the formula:

1 1 1 (z) = p(z, Γ) = z2 + λ Γ 0 (z λ)2 λ2 P ∈ −{ } − − P h i Hence (C) and satisﬁes: P ∈ M (a) (z) = ( z), P P − (b) (z + λ) = (z) for every λ Γ, P P ∈ (c) there exists no other poles.

Hence descends to a meromorphic function on C = C/Γ with a pole at 0 (degree = 2), i.e., we have a holomorphicP function f 1 C CP −→ Locally, this map looks like z zn about 0. In this case n = 2. This map has exactly degree 2 since there are no other poles. Moreover, 7→f is branched at 0.

1 CP ∞

2 Now 0(z) is also periodic (period Γ) with triple poles on Γ and no other poles. The map C P given by P −→ z C [ (z): 0(z) : 1] ∈ 7→ P P deﬁnes a holomorphic function C 0 P2, − { } −→ (z) 1 [ (z): 0(z) : 1] = P : 1 : P P (z) (z) P0 P0 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 a = 0 for every i and P is a discrete subset ofPS. A divisor D is called eﬀective if a 0 for every i. i 6 { i} i ≥

If supp(D) = P / a = 0 is ﬁnite, then { i i 6 }

deg(D) := ai P Deﬁnition 2.5.2. Let f (S) 0 = ∗(S). The divisor of f is deﬁned by ∈ M − { } M

(f) := (f)0 (f) − ∞ where

(f)0 = (ordP f)P and (f) = P f −1( )(multP f)P ∞ ∈ ∞ Note that mult f = ord f, soP we can rewrite the previos expresionP as P − P

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

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

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

1 Example 2.5.1. D1 D2 on P 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 distinct ∼ ∈ 1 points in C and suppose that D = p q = (f) for some f ∗. Then f is a map C P with deg(f) = 1. − ∈ M −→ We have (f)0 = p and f is bijective. Then f is a biholomorphic map, getting a contradiction.

Given ω 0(S)∗. Recall that this means ω = fdz for a local coordinate z at p and f (p). ∈ M ∈ M

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

(ω) := p S(ordpω)P ∈ P is called a canonical divisor and is denoted K or simply K. A divisor is canonical if D (w). S ∼

16 Example 2.5.3.

K = (dz) = 0 P . • C/Γ · 1 dω K 1 = (dz) = 2 , z = , dz = 2 . • P − · ∞ ω − ω

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

p S1 X∈ where each r (f) 0. p ≥

n n n 1 Proof: Locally, f looks like z z , and dz = nz − dz. 7→

Corollary 2.5.1 (Riemann - Hurewicz formula).

2 g(S ) 2 = degK = (degf) degK + degR, · 1 − S1 · S2 where K = (ω), ω Γ(T ∨S ) (global sections of the tangent bundle) ω∨ Γ(TS ). S1 ∈ 1 ←→ ∈ 1

17 2.6 Covering spaces

Recall that an étaléspace (or étalemap) 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 image p(U) is open in X and the restriction of∈p to U is a homeomorphism p : U e p(U). A connected covering |U −→ space p : X X is a universal covere if X is simply connected. Thee name universal cover comes from the following−→ important property: if the map p : X X is a universal cover of the space X and the map −→ p0 : X0 e X is any cover of the space X wheree the covering space X0 is connected, then there exists a −→ covering map f : X0 X such that p f = p0. Anye manifold X has a universal cover X with étalecovering map f : X X and−→ π (X) acts on◦X discretely and freely with quotient f. Moreover, there exists a −→ 1 bijection e e e πe(S) H X/H étale X 1 ⊇ 7→ { −→ } between the set of subgroups of π (X) up to conjugation and the set of étalecoverings from a connected 1 e manifold. G/H for H normal Galois (regular) covering. ⇐⇒ 1 Recall that a covering map p : X X is said to be Galois if for every x X and x p− (x), the subgroup −→ ∈ ∈ p π1(X, x) is normal in π1(X, x). ∗ e e If X ise ae Riemann surface, then the complex charts on X lifts to any covering space.

Lemma 2.6.1. Let f : Z1 Z be a finite étalecovering corresponding to H, π1(Z). Then there exists a finite regular covering h :−→Z Z and a finite étalecovering g : Z Z such→ that f g = h. 2 −→ 2 −→ 1 ◦

Z1 Z2

´etalecovering regular Z ∃

Proof: H has a ﬁnite number of conjugates in π1(Z), such a number equals [π1(Z): N(H)] and these intersection 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 isomorphic n −→ to D∗ D∗ (z z ). −→ 7→

Example 2.6.1. If f : Z Z is finite between Riemann surfaces, then there is a finite étalecovering 1 −→ 2 1 Z f − (∆) Z ∆ 1 − −→ 2 − 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 étalecovering U Z2 ∆, where U is an open subset, has a unique continuation⊆ to a finite map −→ −

U Z Z ⊆ 1 −→ 2 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. Let n n 1 P (T ) = T + c T − + + c 1 ··· n in (Z2)[T ] be an irreducible polynomial. Then there exists a map of Riemann surfaces f : Z1 Z2 of degreeM n and a meromorphic function F (Z ) that satisﬁes: −→ ∈ M 1 n n 1 F + f ∗(c )F − + + f ∗(c ) = 0. ( ) 1 ··· n ∗

Proof: Let ∆ Z be the discrete set containing the poles of the c ’s and the points p where ⊆ 2 i n n 1 P (T ) := T + c (p)T − + + c (p) p 1 ··· n

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

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

Corollary 2.7.1. If Z Z is ﬁnite, then 1 −→ 2

f ∗ : (Z ) (Z ) M 2 −→ M 1 is a ﬁnite ﬁeld extension.

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

Theorem 2.7.1. As soon as there exists a meromorphic function f on Z ( compact = f is ﬁnite), 1 ←− ⇒ (Z1) is ﬁnite algebraic over C(z) of extension deg = degf. M f 1 Z1 P −→

Corollary 2.7.2.

∗ f f (1) If Z Z is finite then (Z ) , (Z ) is a finite field extension of degree (f). 1 −→ 2 M 2 → M 1 ϕ (2) Conversely, let (Z2) , (Z1) = K be a finite field extension of degree d. Then there exists a finite map Z Z Mof degree→d Mwhose field extension is isomorphic to ϕ. 1 −→ 2 (3) A field K of transcendental degree = 1 over C is isomorphic to (Z) of some compact Riemann surface. Such a Z is called a model of K. M

20 2.8 Review

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

ω1 ω , ω 0(Z) = = (f) Div (Z), where f (Z). 1 2 ∈ M ⇒ ω ∈ P ∈ M 2

Here, Div denotes the set of principal divisors. Note that ω 0(Z) if and only if ω Γ (T ∨Z), where P ∈ M ∈ m T ∨Z denotes the cotangent bundle of Z and Γm(T ∨Z) is the set of holomorphic sections of T ∨Z. Also, T T ∨ = (trivial line bundle) and so ⊗ O 1 1 = ω Γ ( ) = , ω · ∈ m O O 1 where TZ and ω T ∨Z. ω ∈ ∈

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). Then df dz df f = n z and p Z Resp f = 0, where Z is compact. Hence we get the following result: ∈ P

Theorem 2.8.1. deg(ω) has the same value for any ω 0(Z), assuming that Z is a compact and connected Riemann surface. ∈ M

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

deg(ω) = 2g 2 and deg 1 = 2 2g = χ(Z) − ω − where 1 Γ(TZ). ω ∈

genus = # of holes

Recall the following results:

21 n n 1 Theorem 2.8.2. Let P (T ) = T + c T − + + c (Z)[T ] be an irreducible polynomial in T over . 1 ··· n ∈ M Z Then there exists a ﬁnite map of Riemann surfaces f : Z0 Z of degree n, unique up to isomorphisms, −→ and a meromorphic function F on Z0 satisfying

n n 1 F + (f ∗c )F − + + (f ∗c ) = 0 1 ··· n

Corollary 2.8.1.

ϕ (1) Z Z finite = f ∗ : (Z ) , (Z ) is a finite field extension of degree n. 1 −→ 2 ⇒ M 2 → M 1 (2) Conversely, any finite field extension of degree n gives rise to a finite map Z1 Z2 whose associated field 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 isomorphic 1 to (Z) for some Riemann surface Z (finite over CP ). M

Z is called a smooth model of K.

2 Theorem 2.8.3. Let C CP be an irreducible plane curve, i.e., C = V (F ) where F is an irreducible ⊆ homogeneous polynomial in (z0 : z1 : z2). Then there exists a compact Riemann surface Z and a generically 2 injective map f : Z CP 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 ﬁeld − K. 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 : Z Z is a finite map, then K f ∗K + R (f ∗ = pullback) where 1 −→ 2 Z1 ∼ Z2 R 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 P deg(f) = ordp(f), p fibre ∈X where this sum is independent of the choice of the fibre.

Corollary 2.8.2 (Riemann-Hurwitz).

deg(K ) = (deg(f)) (deg(K )) + deg(R) Z1 · Z2

Corollary 2.8.3. g(Z ) g(Z ), where g is for genus. 1 ≥ 2

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(Hol0(Z)) where Hol0(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 manifold is orientable if its transition functions have positive Jacobian . Let Z be a Riemann surface, consider two intersecting charts U and U , and let z U U . Then we have α β ∈ α ∩ β df (z) = f 0(z) as an R-matrix. dz We write f = u + iv, then df = α + iβ.

β α 1 α β− ◦ holomorphic C

By the Cauchy-Riemann equations, we have

α β df = β α − and so 1 1 2 2 2 Jac(α β− ) = det(D(α β− )) = df = det(df) = α + β > 0. ◦ ◦ | | 1 It is also known that every Riemann surface is obtained by attaching handles to CP = S2.

# handles = # holes = : topological genus = g

∞

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 attaching 2 1 handles to S = CP = C . We denote Zg for a Riemann surface Z of genus g. It is known that the ∪ {∞} ﬁrst homotopy group of Zg is given by

1 1 1 1 π1(Zg) = Za1 Zb1 Zag Zbg / a1b1a− b− agbga− b− ∗ ∗ · · · ∗ ∗ 1 1 ··· g g

Notice that every hole gives two generators.

b b1 g

ag a1

b1 1 a1− a1 1 b1−

1 bg−

i Let denote the space of C∞-complex valued i-forms on a Riemann surface, then A = 1,0 0,1, A A ⊕ A ω0 = fdx + gdy = hdz + hdz, where hdz 1,0 and hdz 0,1. We also have diﬀerential operators making the following diagram commutes ∈ A ∈ A

1 A d d

0 ∂ 0,1 ∂ 2 C∞ functions = − A A A

∂ ∂ 1,0 A where d = ∂ + ∂.

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

Since d d = 0, we have Im(d) Ker(d), and so we deﬁne the de Rham cohomology groups of Z and the quotient◦ ⊆ Hi (Z) = closed i-forms / exact i-forms dR { } { } 25 0 Note that HdR(Z) = C if Z is connected. If Z is also compact, by the Poincar´eduality Theorem we have an isomorphism H2 (Z) H0 (Z) given by dR −→ dR [ω] ω 7→ ZZ n Also, HdR(Z) = 0 for every n 3. For any compact and connected Riemann surface Zg, the middle cohomology is given by ≥

1 H (Zg) = π1(Zg)/ commutator subgroup = Za1 Zb1 Zag Zbg dR h i ⊕ ⊕ · · · ⊕ ⊕

Theorem 2.9.2. Let ω be a holomorphic diﬀerential ( 1-form). Then ω is d-closed (hence [ω] H1 (Z)). ←→ ∈ dR

Proof: Locally, ω = f(z)dz where f . Then ∈ O ∂f ∂f dω = (∂ + ∂)ω = ∂ω + ∂ω = dz dz + dz dz ∂z ∧ ∂z ∧ = ∂f dz + ∂f d = 0 + 0dz dz, since f is holomorphic. ∧ ∧ ∧

26 i 2.10 Product structures on i HdR(Z)

By taking wedge products of forms, we have a surjectiveL map H1 (Z) H1 (Z) C given by dR × dR −→

([ω ], [ω ]) ω ω 1 2 7→ 1 ∧ 2 ZZ 1 1 In this situation we have HdR(Z) ∼= (HdR(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. ∩ × −→

1 By the Poincar´eduality Theorem we have H (Z) = H1(Z)∨ C. Also dR ∼ ⊗ 1 H (Z) = (H1(Z)∗ = H1(Z)) C. dR ⊗ Recall that the Euler characteristic of Z is given by

χ(Z) := ( 1)idimHi (Z) = ( 1)idim H (Z) = 2 2g i − dR i − Z i − P P 0 2 1 2g Recall HdR(Z) = HdR(Z) = C and HdR(Z) = C .

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

Pg(Z) = dimC(Ω)

Clearly, Ω , H1 (Z) if Z is compact. We have that → dR dg fdz = dz = dg = 0 dz

1 implies that g . Since Z is compact, we get that g = const. Similarly, Ω , HdR(Z). It follows 2Pg 2g. It is diﬃcult to∈ determine O when they are equal. The fact that P 2 implies→ that there exist meromorphic≥ g ≥ functions. If Pg = g, then a loop C on a compact Riemann surface Z is homotopic to 0 if and only if C ω = 0, for every ω Ω. The map ∈ R ω ω 7→ C is called the period map. R

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 consider each a and b as maps S1 S or− curves with parameter t [0, 1] in S, oriented counterclockwise. i i −→ ∈

27 b b1 g

ag a1

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 only if a + ( b) = a b = ∂U, for some open set U S, i.e., a is homologous to b. Here b denotes reverse orientation.− So if−a is homotopic to b then they are⊆ homologous. −

Consider the following picture:

chart

−→b y

p x −→a

We have a −→b = kx y, and denote −→ ∧p ∧p (a, b)p = sign(k) and

(a, b) = p a b(a, b)p ∈ ∩ Since the previous sum does not depend on the choiceP 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, (a , b ) = (b , a ) = δ . i j − j i ij

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

ai bi ai 01 det bi 10 1 −

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

Let ω 0(S) be a closed meromorphic diﬀerential. Recall that ω is exact if and only if ω = df for some meromorphic∈ M function f (S). Can we choose a function f (S) such that f = z ω? As an exercise, ∈ M ∈ M p think if it is possible that Res (ω) = 0 for every p S. Consider the period homomorphism p ∈ R Πω : H1(S; Z) C −→ [γ] ω, γ is a loop. 7→ Zγ

This map is well deﬁned. For if γ = γ0 in H then γ γ0 = ∂u, and by the Stokes Theorem we have 1 − γ ω = γ0 ω. Also, Πω is a C-linear map. R R 1 1 By the de Rham Theorem, we have an isomorphism HdR(Z) ∼= H (S; Z) C if Z is compact. Such an isomorphism is given by ⊗ [ω] H1 (Z) Π ∈ dR 7→ ω

1 Corollary 2.10.1. dimCHdR(S) = 2g.

Proof: We have

2 2g = χ(S) − = ( 1)ihi (S), where hi (S) = dim(H1 (S)), − dR dR dR = Xh0 (S) h1 (S) + h2 (S) dR − dR dR = 1 dim H1 (S) + 1. − C dR

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

1 1 H H C dR × dR −→ ([ω ], [ω ]) ω ω 1 2 7→ 1 ∧ 2 ZS isomorphically to the (cup) product on H1(S).

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

1 ∼= 1 H H (S; Z) C = Hom (H1, C) dR −→ ⊗ C if S is compact, we have an inclusion Ω , H1 if S is compact. We have → dR ω exact = ω = df = f holomorphic = f constant = ω = 0 ⇒ ⇒ ⇒ ⇒

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

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

30 2.11 Questions about (compact) Riemann surfaces

Recall that if f : Z Z is a finite map of degree n of Riemann surfaces, then any meromorphic function 1 −→ 2 on Z1 satisfies a polynomial of degree n over the field of Z2. Hence

f ∗ (Z ) , f ∗(Z ) M 2 → 1 is a ﬁeld extension of degree = n. 6

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

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

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

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

1 If yes, then a loop C is homologous to 0 if and only if C ω = 0 for every ω Ω. Also, HdR = Ω Ω. The values ω and ω are called periods. ∈ ⊕ ai bi R R R

31 2.12 Harmonic diﬀerentials and Hodge decompositions

1 Recall that = C∞-valued 1-forms. A

Definition 2.12.1. Notice that locally every C∞-valued 1-form can be written as ω = fdz + gdz, where fdz 1,0 and gdz 0,1. There exists a C-linear map : 1 1 called the star operator, locally defined∈ A by ∈ A ∗ A −→ A (pdx + qdy) = qdx + pdy ∗ − Or equivalently, (fdz + gdz) = i( fdz + gdz). ∗ − Every ω is uniquely a sum of ω1,0 1,0 and ω0,1 0,1, and ∈ A ∈ A (ω) = i( ω1,0 + ω0,1). ∗ − Definition 2.12.2. ω 1 is harmonic if dω = 0 = d( ω). An 1-form ω is called coclosed if d( ω) = 0. In other words, ω is harmonic∈ A if it is closed and coclosed.∗ ∗

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

Proposition 2.12.1. Let 1 be the space of harmonic diﬀerentials on Z. Then 1 = Ω Ω. H H ⊕

The Hodge Theorem states that 1 = H1 . H dR

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

We have a positive answer to all questions we established.

1 1 Theorem 2.12.1 (Hodge Decomposition). HdR(Z) = = Ω Ω, where the ﬁrst equality is known as the Hodge Theorem. H ⊕

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 ∈ ≥ − { } 1 n ω d = ω + dz − zn zn+1 is harmonic on a neighbourhood U of p, and ω B , i.e., ω ω < . S0 U S U ∈ − − ∧ ∞ R

32 Corollary 2.12.1.

(1) There exist meromorphic differentials on Z with any preassigned finite set of poles p and any principal parts ∞ ω = a zidz, when n 2. p i ≥ i= n X− (2) There exist a meromorphic function with any prescribed value at a finite set of points.

(3) f ∗ , has degree = deg(f) for a ﬁnite map f : Z Z . M2 → M1 1 −→ 2

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)

(ω , ω ) = ω ω < . 1 2 Z 1 ∧ ∗ 2 ∞ R Locally, ωi = pidx + gidy, with i = 1, 2. Then

ω ω = (p p + g g )dx dy. 1 ∧ ∗ 2 1 2 1 2 ∧ Hence we can deﬁne 2 ω 2 = ω ω < . || ||L ∧ ∗ ∞ ZZ 2 Deﬁnition 2.13.1. Let B0 be the space of bounded 1-forms ω such that ω < . || || ∞

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

1 0 0 Let E be the closure in B of d , where A is the space of C∞-functions with compact support. AC C

Theorem 2.13.1 (Orthogonal decomposition). Let ω 1. Then there exists a unique orthogonal decomposition ∈ B ω = ω + df + dg h ∗ where ω is bounded harmonic and f, g 0, and df, dg E. h ∈ A ∈

Proof: The essential point is that the space is orthogonal to both E and E, and E E. If ψ, ϕ 0 then H ∗ ⊥ ∗ ∈ A dϕ, dψ = dϕ dψ = ψddϕ + d(ψdϕ) ∗ − ∧ ZZ ZZ ZZ = 0 + 0.

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

0 = d ω, ϕ = d ω ϕ = dϕ ω = dϕ, ω h ∗ i ∗ ∧ ∧ ∗ h i Z Z = ω, dϕ . h i Hence E E, B1. H ⊕⊥ ⊕⊥ ∗ → 2 To show the equality, we go to the L -completion of B0 ﬁrst.

1 2 Theorem 2.13.2 (Regularity). = (Eˇ Eˇ)⊥ in Bˇ where (∨) means L -completion. H ⊕ ∗

34 Corollary 2.13.1 (Hodge decomposition). H1 (Z) = Ω Ω. dR ⊕

Proof: Saying that a form is closed means that it is orthogonal to E. Hence the previous theorem ∗ 1 implies that a closed 1-form ω is uniquely ωh + df, i.e., every element of HdR has a unique harmonic representative.

35 2.14 Review

Theorem 2.14.1 (Orthogonal decomposition). For every ω 1 = space of diﬀerntial 1-forms, there exists a unique decomposition ∈ A ω = ω + df + dg h ∗ where ω is a bounded harmonic, f, g 0 = space of smooth functions. h ∈ A

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

Proof: Let be the space of harmonic diﬀerentials. Then is orthogonal to both E = d 0 and E = d 0.H This fact follows easily using the L2-inner productH and the equality = ( 1)Ak, for ∗Riemann∗ A surfaces one has ( 1)k = 1. We have ∗∗ − − E E 1. H⊥ ⊥ ∗ ⊆ A Taking completion, we have ˇ Eˇ Eˇ = 1 H ⊥ ∗ A where ˇ Eˇ is the orthogonal ( ˇ Eˇ) direct sumM of ˇ and Eˇ. By the Weyl’s Lemma, we have = ˇ. H ⊥ H ⊥ H H H

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

Tf [h] = hf ZU and h is compactly supported in U Z, where Z is a Riemann surface. ⊂⊂

Corollary 2.14.1 (Hodge decomposition). H1 (Z) = Ω Ω = . dR ⊕ H

36 2.15 Proof of Weyl’s Lemma

Claim 2.15.1. If f 0(C) then there exists ψ 0(C) such that ∆ψ = f. ∈ A ∈ A

0 Deﬁnition 2.15.1. (U) = f C∞(U)/ f has compact support in U . Ac { ∈ }

Note that 0(U) is a topological space of uniform convergence (not complete). Ac

A distribution on U is a continuous linear functional T : 0(U) C. Ac −→

Example 2.15.1. Let h 0. Deﬁne ∈ Ac Th[f] = hf ZU Then T is a distribution. Using integration by parts, for h 0 and f 0, we have ∈ Ac ∈ Ac α α α hD f = ( 1)| | fD h, − Z ZZ α ∂(α1+α2) where α = (α , α ), D = α α and α = α + α . In other words, we have 1 2 ∂x 1 ∂y 2 | | 1 2 α α T α [f] = ( 1)| |T [D f] D h − h α α α 0 Deﬁnition 2.15.2. (D T )[f] = ( 1)| |T [D f], for every f . − ∈ Ac

α 0 The deﬁnition for D Th is the same as TDαh. Let g (U I) where I is an interval, supp(g) K I and K U. ∈ A × ⊆ × ⊂⊂

g(z,t+) g(z,t) ∂g Let f (z) = − . Then f as 0. By continuity of T , we get −→ ∂t −→

d 0 0 ∂g(z, t) T (g(z, t)) → T [f ] → T dt z ←− −→ z ∂t So:

d ∂g(z,t) Claim 2.15.2. dt Tz[g(z, t)] = Tz ∂t . h i

Similarly, if U and V are open in C and K U, L V , g 0(U V ) are such that supp(g) K L, then ⊂⊂ ⊂⊂ ∈ A × ⊆ ×

Tz g(z, )d(Vol) = Tz(g(z, ))d ZV ZV 0 1 z Let ρ ( ) satisfy supp(ρ) , ρ(z) = ρ0( z ) for every z, and ρ = 1. Set ρ(z) = 2 ρ . Then C D C f 0∈(U A) implies ⊆ | | ∈ A R (ρ f)(z) = ρ (z )f()dVol() 0(U ()) ∗ − ∈ A ZU where U () = z U / d(z, ∂U) > . { ∈ }

37 Claim 2.15.3. For every α N2, we have ∈ Dα(ρ ρ) = ρ (Dαf) ∗ ∗

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

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

Proof:

2π (ρ f)(z) = ρ ()f(z + )dVol() = ρ (r)f(z + riθ)rdrdθ = 2πf(z) ρ (r)rdr = f(z). ∗ ZD(0,z) Z0 Z0 Z0

Proof of Weyl’s Lemma: Since ρ ( z) has compact support in U, for every z U (), −→ − ∈ h(z) := T [ρ ( z)] − is deﬁned and belongs to 0(U ()) by Claim 6.2. By Claim 6.4, it suﬃces to show that for every f 0(U ()) we have A ∈ Ac T [f] = hf, () ZU where h is harmonic since ∆h = T [∆ ρ ( z)] = ∆T [ρ( z)] = 0. z − − We want to show that T = Th. We have

T [ρ f] = hf. ∗ () ZU Hence it suﬃces to show T [f] = T [ρ f]. ∗ By Claim 6.1, there exists ψ 0 such that ∆ψ = f, where ψ is harmonic on V = C supp(f). Hence ψ = ρ ψ on V by Claim 6.4.∈ A Therefore, = ψ ρ ρ 0(U) satisﬁes − ∗ O − ∗ ∈ Ac ∆ = ∆ψ ρ ∆ψ = f ρ f O − ∗ − ∗ since ∆T = 0, and T (∆ψ) = 0. Hence T [f] = T [ρ f + ∆ ] = T [ρ f]. ∗ O ∗

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 there exists a harmonic diﬀerntial ω on Z p such that: ∈ ≥ − { } 1 (1) ω d n is harmonic on a small neighbourhood of p. − z 2 (2) ω B , i.e., ω is bounded and smooth outside U, with ω 2 < . Z0 U L ∈ − || || ∞

Outline of the proof: Let ρ(z) be a diﬀerentiable function on Z such that ρ 0 outside U and ρ 1 on a neighbourhood of p. The form ≡ ≡ ρ(z) ψ = d (p). zn ∈ M We take p = 0. Now ψ i (ψ) is smooth and has compact support on Z ( 0 on a neighbourhood of 0 and outside U), where −(ψ∗) = i(udz vdz) if ψ = udz + vdz. Hence by the≡ Decomposition Theorem we have ∗ − ψ i (ψ) = ω + df + (dg) − ∗ h ∗ and ω = ψ df = ω + i (ψ) + (dg) − h ∗ ∗ 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 1 such that dh < and dh 0 on a neighbourhood of p. ∈ A || || ∞ ≡ This condition is the same as the following: Since dh 0 on a neighbourhood N of p, we have ≡ 2 2 ω + dh Z N = ω Z N + (dh, dh) + (ω, dh) + (ω, dh) || || − || || − 2 2 = ω Z N + dh Z N || || − || || − 2 ω Z N . ≥ || || −

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

n Proof: f = (1, f1, f2, . . . , fn): Z CP if f(P ) = f(Q) for P = Q. Then just adding a meromorphic function that separates P and Q.−→ This process must terminate by6 compactness.

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

2 3 z0 z1 Proof: Case n = 2: Consider f : Z CP and [z0, z1, z2] C . Then a = f ∗ and b = f ∗ −→ ∈ z2 z2 are algebraic dependent over C, i.e., there exists a polynomial F (Z1,Z2) of degree dsuch that F (a, b) = 0. Then xdF x0 , x1 deﬁnes f(Z). 2 x2 x2

This result is also known as the 1-st GAGA Principle (after GéométrieAlgébriqueet GéométrieAnalytique).

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 holomorphic 1 map Z CP with 3 ramiﬁcation points. −→

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

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

1 Corollary 2.18.1. A Riemann surface Z is rational (∼= CP ) 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 (x) for each x M, given by I ∈ (x) = f C∞(M)/ f(x) = 0 , C∞(M) I { ∈ } → The cotangent plane at x M can be deﬁned as the quotient ∈ 2 T ∨(M) := (x)/ (x) x I I and the tangent plane at x M is simply the dual space of the cotangent plane T ∨(X), i.e., ∈ x

Tx(M) := (Tx∨(M))∨

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

A complex structure on X induces an almost complex structure on X by setting J = i = √ 1. We obtain a map J : T T with √ 1 acting on the domain and J acting in the codomain. We− have X,R −→ X,R − ∂ 1 ∂ ∂ ∂ ∂ = i , ∂z 2 ∂x − ∂y 7→ ∂x ∂y i i i i Locally, J is deﬁned by ∂ ∂ ∂ ∂ , , ∂x ∂y 7→ ∂y −∂x i i i i 1,0 Let (X,J) be an almost complex manifold. Then TX,R C contains an eigen-bundle TX corresponding to 0,1 ⊗ the eigen-value i, an an eigen-value TX corresponding to the eigen-value i, for the operator J. Note that 1,0 − TX is naturally isomorphic to TX,R by taking the real part, and this isomorphism identiﬁes i with J. Hence T 1,0 is generated by vectors of the form u iJu, with u T . X − ∈ X,R

Theorem 3.1.1. A complex manifold has a complex structure J on TX,R and its associated subbundle 1,0 T TX, C is naturally the same as TX (by taking the real part). X ⊆ R ⊗

43 Similarly, if Ω := T , then X,R X,∨ R ΩX, C = ΩX ΩX R ⊗ ⊕ with the identiﬁcation Ω Ω1,0 dz , X ←→ X ←→ i Ω Ω0,1 dz . X ←→ X ←→ i

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 complex structure via the octonions, by taking the multiplication structure in the multiplicative structure in the sphere 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 1,0,T 1,0] T 1,0. X X ⊆ X

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

Proposition 3.1.2 (∂-PoincaréLemma). Let α be a ∂-closed differential from a differentiable manifold with (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 = α dzI dzJ I,J ∧ ∂α = X dα dzI dzJ = 0 I,J ∧ ∧ J X p I Then αI = ∂z is ∂-closed and of type (0, q), and so αI = ∂β if and only if α = ( 1) ∂( ∂z βI ). αI,J − ∧ Henceforth assume that α is of type (0, q), i.e., α = α dzJ . Apply the induction on the largest P J P integer k such that k J and αJ = 0. Necessarily k q and k = q implies α = fdz1 dzq. In the ∈ 6 ≥ P ∧ · · · ∧ latter case ∂α = 0 if and only if f is holomorphic in the variables zi with i > q. We may know apply the following result:

Proposition: There exists a diﬀerentiable function g holomorphic in the variables zi, with i > q, such ∂g q 1 that = f and hence α = ( 1) − ∂(gdz1 dzq 1). ∂z q − ∧ · · · ∧ −

Now assume the ∂-Poincar´eLemma proved for k 1 > q. Write α = α + α dz , α = α dzJ where − 1 2 k 2 2,J J = q 1 and J 1, . . . , k 1 . So ∂ = 0 implies α2,J is holomorphic in variables zl, with l > k. | | − ⊆ { − } P Hence by the previous proposition, we have α = ∂β2,J , where β is holomorphic in z , l > k. Then 2,J ∂zk 2,J l

J q 1 ∂β = ∂(β dz ) = ( 1) − α dz + α0 2,J − 2 ∧ k 1

44 where α10 involves only the coordinate zl for l < k. Thus,

α = α100 + ∂β where α100 for l < k. Since β is holomorphic in the zl for l > k, we have ∂α100 = 0, q = deg(α100) < k. We conclude by induction.

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

1 f(z) 1 f(z) g = dζ dζ := lim dz dz. 2πi ζ z ∧ 0 2πi D(z,) z ∧ ZC − → ZC\ − 1 This limit exists since f is bounded and ζ z in integrable on D. We want to show that ∂g = α = fdz. − 1 f(γ+z) Now g = g where g = 2πi D(0,) γ dγ dγ. Then C\ ∧ R 1 dγ dγ ∂g(z) = ∂zf(γ + z) ∧ dz 2πi D(0,) γ ZC\ ! implies ∂ 1 ∂ dγ dγ g(z) = ∂ g(z) = f(γ + z) ∧ ∂z z 2πi ∂z γ ZC 1 ∂ dζ dζ = lim f(ζ) ∧ . 0 2πi D(z,) ∂ζ ζ z → ZC\ −

∂f dζ dζ dζ On D(z, ), we have (z) ∧ = d f(ζ) . By the Stokes Theorem, we get ∂ζ ζ z ζ z C\ − − − 1 ∂f dζ dζ 1 f(ζ) ∧ = f(z) as 0. 2πi D(z,) ∂ζ ζ z 2πi ∂D(z,) ζ z −→ −→ ZC\ − Z − Hence ∂g = fdz.

45 3.2 K¨ahlermanifolds

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

Deﬁnition 3.2.1. Let W 1,1 = W 1,0 W 0,1 ∆2W ∆2W , ⊗ ⊆ C ⊇ W 1,1 = (1, 1)-forms = sesqui-linear forms on V . { } { } Let W 1,1 = W 1,1 ∆2W = real (1, 1)-forms = real 2-forms of type (1, 1) = alternating forms .A R 1∩,1 { } { } { 1,1 } (1, 1)-form h W is called Hermitian if h(u, v) = h(v, u) for every u, v V . Let WH be the space of such forms. ∈ ∈

Fact 3.2.1. There exists a bijective correspondence between Hermitian forms and real alternating forms of type (1, 1) via W 1,1 h Im(h) W 1,1 H 3 ←→ ∈ 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 and set ∈ ∈ R 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 = i a dz dz = Im(h) Ω1,1 Ω2 , where (a ) is hermitian. 2 ij i ∧ j − ∈ X ∩ X,R ij P Deﬁnition 3.2.2. ω W 1,1 is positive if the correspondence h is positive deﬁnite. ∈ R

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

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

Deﬁnition 3.2.5. Let h be a Hermitian metric. We shall say that h is K¨ahler 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 vanishes on Ω2,0 and hence also on Ω0,2 and its associated h is positive deﬁnite), then it is Im(h) for a K¨ahlermetric. −

Corollary 3.2.3. A Hermitian metric can always be written as h = g + iω where g is a Riemannian metric invariant 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 a K¨ahlermanifold.

n Lemma 3.2.1. dVol = ω for (X, h), h = h = g(u, v), where ωn = ω ω of type (n, n). h n! ω ∧ · · · ∧

Proof: Let ei be an orthonormal basis of TX,x with respect to h. Then ei, Jei is a real orthonormal { } { n } basis for T R with respect to g with positive orientation. It suﬃces to check ω = dx dy dx dy X,x n! 1∧ 1∧· · ·∧ n∧ n where dx1, dy1, . . . , dxn, dyn is the dual basis to (ei, Jei). Let dzj = dxj + dyj. Then we have ωx = i dz{ dz and } 2 j j ∧ j P ωn i n x = dz dz at x, n! 2 j ∧ j j Y i dz dz = dx dy . 2 j ∧ j i ∧ j

Corollary 3.2.4. If X(n) is a compact K¨ahlermanifold then [ωk] H2k (X) is nonzero for every k < n. ∈ dR

k n n k n Proof: ω = dγ implies ω = d(ω − γ). The last implies 0 = ω = 0, getting a contradiction. ∧ 6 X R

Corollary 3.2.5. Let X(k) be a compact K¨ahlersubmanifold 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| →

h h h Volume = i∗ω = dω = 0 since dω 0. X M M M ≡ ZX ZΓ

47 3.3 Metrics and connections

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

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

: 0(E) 1(E) ∇ A −→ A satisfying the Leibniz rule: (fσ) = df σ + f σ. ∇ ⊗ ∇ For a vector ﬁeld ψ and σ 0(E) we write ∈ A σ = ( σ)(ψ) 0(E). ∇ψ ∇ ∈ A In the case where E is a holomorphic vector bundle, we have the operation

∂ : 0(E) 0,1(E) E A −→ A 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 unique connection on TM called the Levi-Civita connection satisfying: ∇ (1) d(g(ψ , ψ )) = g(ψ , ψ ) + g( ψ , ψ ), i.e., g is -invariant. 1 2 1 ∇ 2 ∇ 1 2 ∇ (2) ψ ψ = [ψ , ψ ], i.e., g is torsion free of . ∇ψ1 2 − ∇ψ2 1 1 2 ∇

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 1(E) 0,1(E). Then 0,1 = ∂ . ∇ A −→ A ∇ E

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

(1) h is K¨ahler. (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 J 2 = 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) and Hermitian metrics. Such a correspondence is given by

1,1 1,1 W ∼ W H −→ R h Im(h) 7→ 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 K¨ahlermetric, i.e., dwh = 0. (ii) J is ﬂat for the Levi-Civita connection of h.

1,0 R (iii) The Chern connection of h on TM equals the Levi-Civita connection on TM .

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(ϕ , ϕ ) = ω( ϕ , ϕ ) + ω(ϕ , ϕ ). 1 2 ∇ 1 2 1 ∇ 2

Let C∞(M) ϕ[ω(ϕ , ϕ )] = ω( ϕ , ϕ ) + ω(ϕ , ϕ ). Since 3 1 2 ∇ϕ 1 2 1 ∇ϕ 2 dω(ϕ, ϕ , ϕ ) = ϕω(ϕ , ϕ ) ϕ ω(ϕ, ϕ ) + ϕ ω(ϕ , ϕ) ω([ϕ, ϕ ], ϕ ) 1 2 1 2 − 1 2 2 1 − 1 2 the result follows from [ϕ , ϕ ] = ϕ ϕ . i j ∇ϕ1 j − ∇ϕ2 1

(i) = (iii): The Chern connction equals the Levi-Civita connection for the ﬂat metric i dzi dzi. • The⇒ result follows from the following proposition. ∧ P

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

∂ ∂ 2 hij = h , = Im + O( zi ). ∂zi ∂zj | | X The converse is also true.

49 3.5 The Fubini Study metric

n Let L = (l, v) CP Cn / v l . Consider the diagram { ∈ × ∈ } π i n 2 L CP Cn+1 Cn+1 × π i π 1 ◦ 1 n CP

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

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

2 n+1 The standard metric zi on CP restricts to a Hermitian metric on L. Its curvature (Ricci or Chern form) is given by | | P 2 i i 2 ω = σ∗ ∂∂log z = ∂∂log σ 2π | i| 2π | | n for any choice of a holomorphic section σ of L over CP . Therefore, σ0 = σf, for f and so ∈ O 2 2 2 log(σ0) = log σ + log f , | | | | 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

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

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

i 1 π˜ L π− (E) E

Π π P(E) X

where P(E) = (E zero sections )/C∗ and L is denoted by L = OP(E)( 1). The composition π i is called \{ } 1 − ◦ the blow up of E at its zero section. Here π− (E) is the fibre product or pullback of π and Π. Let 2 F = (PE)x X be the fibre of π at x and f : F, PE the inclusion. Then f ∗c1( h) is a positive (1, 1)-form, ∈2 i 2 2 → | | where c1( ) = ∂∂log . Hence c1( ) is a (1, 1)-form on P(E) that is positive in the vertical direction | |h 2π | |h | |h of π of X, where X is Kählerwith Kählerform ωX . Hence P(E) is also Kähler.

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

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

ϕ ϕ Consider the compactiﬁcation E = P(E O) E, E X and E O X are vector bundles over X, ⊕ ⊇ −→ ⊕ −→ 1 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− (E)) and hence it is K¨ahler. ⊕

51 52 Chapter 4

SHEAF COHOMOLOGY

4.1 Sheaves

Deﬁnition 4.1.1. Let X be a topological space and an abelian category. A presheaf on X is a collection of objects (U) of objects in , for each openA subset U X, and a collection of morphismsF F A ⊆ ρ : (U) (V ) UV F −→ F σ σ = ρ (σ) 7→ |V UV for each inclusion of open subsets V, U such that → ρ = ρ ρ . UV VW ◦ UV The last equality is known as compatibility.

A presheaf is called a sheaf if it is saturated, i.e., if it satisﬁes the following condition: Let si (Ui) be a collectionF of sections such that ∈ F s = s , i|Uij j|Uij where U = U U , then there exists a unique section s ( U ) such that s = s . ij i ∩ j ∈ F ∪ i |Ui i

Deﬁnition 4.1.2. A morphism of (pre)sheaves is a map ϕ : which associates to each open subset U X a morphism F −→ G ⊆ ϕ : (U) (U) U F −→ G such that for every V U open ⊆ ρ ϕ = ϕ ρ . (compatibility) UV ◦ U V ◦ UV h ω Example 4.1.1. Sheaves of sections of vector bundles (C∞, C , C for real analytic, , etc). O

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

F G ϕ ! ∃

Fsh

If X is a topological space, its structure sheaf 0 associated to each open subset U is given by the space CX of continuous functions on U. If X is an algebraic variety, its structure sheaf X is given by the space of regular functions on (Zariski) open subsets. A complex algebraic manifold is alsoO a complex manifold, and we write alg and hol to distinguish the structures: OX OX alg Zariski topology, OX −→ hol ordinary topology. OX −→ Deﬁnition 4.1.3. Let be the sheaf of rings over X. A sheaf is called an -module if for every open set U, (U) is a moduleA over (U), compatible with the restrictionF maps. A F A

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

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

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

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

Deﬁnition 4.1.5. Let (f, f #):(X, ) (Y, ) be a morphism of ringed spaces, i.e., f : X Y is a continuous function, and A −→ B −→ # 1 f (V ) (f − (V )) : B −→ A is a morphism for every open subset V compatible with restrictions. The pullback sheaf f ∗ of a -module ( ) G B is deﬁned as follows: set f ∗ (U) = lim (V ) and then set f ∗ be the sheaf associated to the G G f(U), V G G ( ) −→ → presheaf f ∗ f (∗) . G ⊗ B A

Example 4.1.2. The sheaf of holomorphic sections of a holomorphic vector bundles F over X, F i : x , X X { } → ( ) (1) ix∗ =: (the stalk of at x and its elements are germs of sections of F at x). F Fx F 54 (2) i∗ = Fx has ﬁnite dimension over C. xF

Deﬁnition 4.1.6. A sheaf of modules on an algebraic variety (X, X ) is said to be (quasi)-coherent if it is locally isomorphic to the cokernelM of a morphism of free sheavesO (of ﬁnite rank).

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

Deﬁnition 4.1.7. A short sequence of sheaves

0 0 −→ F −→ G −→ H −→ is said to be exact if and only if it is exact on the level of stalks.

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

Example 4.1.3. Let be the sheaf of holomorphic sections of a vector bundle F and i : x (X, ). F { } −→ O ( ) (1) i ∗ =: is called the stalk of i, and it equals the set of germs of sections of F . F Fx (2) i∗ = F , the ﬁbre of F at x, is a ﬁnite dimensional vector space. F x

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

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

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

Example 4.1.4.

alg (1) f : C 0 . Then f = C[z] which is not ﬁnite dimensional. −→ { } ∗OC alg 1 (2) i : C∗ C. Then i ∗ = C[z, z− ] is not of ﬁnite type over C[z]. −→ ∗OC

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

N X = zeroes of f1, . . . , fl over C . { } We have K[X] := C[z1, . . . , zN ]/ f1, . . . , fl h i Then U M (U) is an -module and this correspondence preserves tensor product, exactness, 7→ ⊗K[X] OX OX etc. Call this X -module M˜ . In part, M˜ is quasi-coherent (and coherent if M is of ﬁnite type) and any quasi-coherent O -module is of this form. OX

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

We normally assume that the ground field K is algebraically closed. Then the Nullstellensatz tells us that V ( ) := sup( X / ) X is nonempty if and only if = X . Hence the ringed space (V ( ), X / ) is identifiedI withO theIsubscheme⊆ of X corresponding to I. 6 AO ringed space locally isomorphic toI subschemesO I of affine spaces is called al algebraic scheme. I

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

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

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

Special case: Given a coherent sheaf of X -modules , let = Sym X . Then the associated ﬁbre space O F A O F is called the vector ﬁbre space, denoted by π : V( ) X. Here, Sym means the symmetric product k F −→ k 0(Sym ). Note that ⊕ ≥ F V( )x = ( x/ x x)∨ F F M F 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[]/2).

– The Zariski tangent space: Let x X be a point of al algebraic variety. Then TxX := 2 ∈ ( / )∨. Mx Mx – The normal bundle of Y, X (algebraic subvariety): Let be the ideal sheaf of Y , then 2 → I 2 / = X Y . The normal vector bundle (or normal bundle) of Y in X is / . It is denoted I I I ⊗πO O I I by NY X ( Y ). | −→ 56 – The normal cone of Y in X is deﬁned by

k k+1 CY X := SpecmY k 0 / NY X | ⊕ ≥ I I |

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 T : TX →TY .× And it coincides with d f : T X T Y ,−→ for every×x X×. −→ × f −→ x x −→ f(x) ∈ – We say that x X is a smooth point if Cx(X) = TxX, and X is smooth (non-singular) if all points are. ∈ (II) The cotangent sheaf to X is deﬁned as the conormal sheaf to the diagonal in X X. Its local sections are local forms on TX and such a form d gives a map ×

2 / ∼ Ω0 (x) := Ω0 /(Ω0 ) Mx Mx −→ X X,x X,x ⊗ Mx

where Ω0 = diﬀerential on . X { OX } (III) Blowing up a subscheme: Let , X be an ideal sheaf deﬁning a subscheme Y, X, = k I → O → A k>0 . Then σ : X˜ = proj( ) X is called the blow up of X along Y , where proj( ) = ⊕ I A −→ 1 A Specm(homogeneous decomposition of ). By functoriality, σ− (Y ) is the projection of the algebra k k+1 1 A X Y = k 0 / , i.e., σ− (Y ) Y is the projectivization of the normal cone CY N , i.e., 0A = ⊗O O. ⊕ ≥ I I −→ | ⊗OX

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

Fact 4.1.2.

Any torsion free -module admits a resolution, i.e., a birational morphism σ : Y X such that • OX F −→ σ∗ is locally free. F (Hironaka) Any variety X (any rational map X Y ) admits a resolution of singularities by repeatedly • blow ups along smooth centres (i.e., smooth subvariety)−→

σ X˜ X

57 4.2 Cohomology of sheaves

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

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

I = (jI ) ( UI ) (extension by zero outside UI ) F ∗ F| k k k+1 Deﬁne := I =k+i I and d : by F ⊕| | F F −→ F i ˆ (dσ)j0...jk+1 = ( 1) σj0 ... ji . . . jk+1 U UI − | ∩ i X where j j j , σ = (σ ), σ (U) and I = k + 1. Lastly, we deﬁne ι : 0 by 0 ≤ 1 ≤ · · · ≤ k+1 I I ∈ FI | | F −→ F

ι(σ)i = σ U Ui for σ (U). | ∩ ∈ F

Proposition 4.2.1. This is a resolution.

k (2) de Rham resolution: Let be the sheaf of C∞ (R or -valued) differential forms of degree k (on a real or complex manifold).A The d-PoincaréLemma saysC that the complex ( k, d) is a resolution of A Ker(d0) = R (resp. C), constant sheaves over X. (3) Dolbeault resolution: Let E be a holomorphic vector bundle over a complex manifold X and its 0,q 0,q E sheaf of holomorphic sections (i.e., = X (E)). Let be the sheaf of C∞-sections of ΩX E. E O 0,q A ⊗ Generalizing the d-PoincaréLemma, we get that ( (E), ∂) is a resolution of Ker(∂0) = = X (E) (a coherent -module). A E O OX

58 4.3 Coherent sheaves

Let X be the space of functions on X. The “sheaf version”of this space means that to every open subset U OX 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 poles on U. To any complex manifold X we associate the space of holomorphic functions on U.

n A coherent sheaf is free if it is of the form X⊕ . An ideal sheaf is just a subsheaf of X which is an ideal of (U) (a ring) for every U. Normally assumeO = . We haveI a short exact sequenceO OX I 6 OX

0 X Z( ) 0, −→ I −→ O −→ O I −→ where Z( ) is a scheme equal to Z( ) = spec( X / ) ( X, which is nonempty. Note that Z( ) and are I I O I O I I examples of coherent sheaves over X, and Z( ) is called the torsion part. Any coherent sheaf is locally a O I ﬁnite direct sum of these factors. If there are no factors of the form Z( ), i.e., not supported on a proper O I subvariety, then it is called torsion free. Any torsion free sheaf admits a resolution f : X res X such −→ that f ∗ of the sheaf is locally free (i.e., a vector bundle). This is because given an ideal sheaf deﬁning a k I subscheme Y X, A = k>0 is an algebra over X . Then σ : X = proj(A) X (ane isomorphism outside Y ), called⊆ the blowup⊕ Iof X along Y , is a birationalO map to X that replaces−→Y by a subscheme of 1 1 codimension 1, i.e., σ− (Y ) is locally given by one equation and σ− (Y ) Y is the projectivization of the −→ normal cone CY X . |

i i i+1 Given a collection of sheaves over X with morphism di : such that di+1 di = 0 for every i. It is a resolution of a sheafF if there exists an inclusionFi −→: F such that j( ◦) = Ker(d ) and F F −→ F0 F 0 Ker(di+1) = Im(di) for every i.

(1) There exists a sheaf X, Ui i N covering of X. For every ﬁnite set I N, set UI = i I Ui, jI : UI , X, { } ∈ k ⊆ ∩k∈ k+1 → and I = (jI ) ( UI ) extended by zero outside UI . Set = I =k+1 I and d : by F ∗ F F ⊕| | F F −→ F (dσ) = ( 1)i(σ ) , j0 jk+1 j0 jˆi jk+1 U UI ··· − ··· ··· | ∩ i X k 0 where σ = (σI ), σI I (U), and j : is given by j(σ)i = σ Ui U for σ (U). ∈ F F −→ F | ∩ ∈ F

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

k (2) de Rham resolution: Let be the sheaf of C∞ (R or C)-valued k-forms. The Poincar´eLemma k A says that the complex ( , d) is a resolution of Ker(d0) = R or C, constant sheaves. A (3) Let E be a holomorphic vector bundle and its sheaf of holomorphic sections. Let 0,q(E) be the 0,q E 0,q A sheaf of C∞-sections ΩX E. The ∂-Poincar´eLemma implies that ( (E), ∂) is a resolution of Ker(∂ ) = . ⊗ A 0 E

59 4.4 Derived functors

Given a functor F : 0 between abelian categories such that has enough injectives. C −→ C C

i Deﬁnition 4.4.1. For every object M , there exists an object R F (M) for every i 0 in 0, unique up to isomorphisms, satisfying the following∈ conditions: C ≥ C

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

(2) Every short exact sequence 0 A B C 0 gives rise to a long exact sequence in 0 −→ −→ −→ −→ C 0 F (A) F (B) F (C) R1F (A) −→ −→ −→ −→ −→ · · · RiF (A) RiF (B) RiF (C) Ri+1F (A) · · · −→ −→ −→ −→ −→ · · ·

Remark 4.4.1.

(1) Let 0, i : 0 be an acyclic resolution of (i.e., Rj+1F ( k) = 0 for every j, k 0). Then M A −→ M A M ≥ j j 0 i 1 i 1 i R F (A) = H (F ( )) := CoKer(d − ( − ) ( )) M F M −→ F M (2) For a sheaf over X there exists an acyclic resolution (called the Godement resolution for ). F F

Definition 4.4.2. A fine sheaf os a sheaf of -modules where the sheaf of algebra admits a partition of unity: for every open covering FU , there existsAf (U ) such that f = 1 (thisA sum is locally finite). { i} i ∈ F i i i P Proposition 4.4.1. Hi(X, ) = 0 for every i 0 for such . F ≥ F

Corollary 4.4.1.

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

Ker(d) H∗(X; R) = = H∗ (X; R), Im(d) dR

where the same equality holds if we replace R by C. (2) Given a holomorphic vector bundle E X and its sheaf of holomorphic sections. Then −→ E Ker(∂ : 0,q(E) 0,q+1(E)) H∗(X, ) = A −→ A E Im(∂ : 0,q 1(E) 0,q(E)) A − −→ A

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

Consider a Cechˇ resolution Ker(d) = 0 d 1 k F −→ F −→ F −→ · · · −→ F −→ · · · 0 k where the map is given by σ σ U Ui and = I =k+1(jI ) ( UI ). F −→ F 7→ | ∩ F ⊕| | ∗ F|

60 q>0 Theorem 4.4.1. If H (UI , ) = 0 for every I N then F ⊆ Hq(X, ) = Hˇ q(U, ) := Hq(Γ( ) = (X), d ), F F F F X e.g., if X is a ﬁnite dimensional manifold C∞, then there exists a good cover (UI contractible for every I) and so ˇ q q H (U, Z) = Hcell(X, Z) q where U is in an open cover and Hcell(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 k k ( , ) on ∆∗T ∨ . We assume X is oriented. Let α, β : C∞( T ∨). Then X,x ∈ A A X

α, β = α, β dVol(x) h i h ix ZX 2 k n k k n gives an L -metric on . We also have a pointwise isomorphism p : ∆ − Tx∨ ∼ Hom(∆ Tx∨, ∆ Tx∨) given A n k −→ k by v v , where ∆ T ∨ = RdVol(x), and an isomorphism m : ∆ T ∨ ∼ Hom(∆ T ∨, R) given by 7→ ∧ − x x −→ e e, k ∨ . 7→ h i∆ T

Deﬁnition 5.1.1. The Hodge Star Operator is given by

1 k n k = p− m : ∆ T ∨ ∼ ∆ − T ∨ ∗ ◦ x −→ x and the associated global isomorphism by

k n k : ∆ T ∨ ∼ ∆ − T ∨ ∗ −→ k n k Ω (X) Ω − (X) −→

We extend to complex-valued forms by extending , to Hermit metrics on ∆k (T ) = (∆k T ) . C ∨ C R ∨ C We get (α, β∗) dVol(x) = α β and so h i ⊗ ⊗ x x ∧ x

α, β = α β h i ∧ ∗ ZX 2 k k k p,q p,q p,q is the -metric on = C. In the case X is complex, = p+q=k , = C∞(Ω ). L AC A ⊗ AC ⊕ ⊕ A A X

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

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

2 k Fact 5.1.2. ∂∗ = ∂ and ∂∗ = ∂ are formal adjoint of ∂ and ∂ with respect to the metric in . − ∗ ∗ − ∗ ∗ L AC

α α 1 Proof: (∂α, β) = ∂α β = ( 1)| |α ∂ β = ( 1)| |α ∂ β = (α, ∂∗β). X ∧ ∗ − X − ∧ ∗ − X − ∧ ∗ ∗− ∗ R R R

More generally, if (E, h) is a Hermitian vector bundle then there exists a C-anti linear isomorphism of vector 0,q 0,q 0,q n,n q 2n n,n bundles given by h :ΩX E = Ω (E) (Ω E)∨ ∼= Ω − E∨, where ∆X = Ω = RdVol(x). So it gives am antilinear isomorphism⊗ −→ ⊗ ⊗

0,q n,n q 0,n q :Ω (E) ∼ Ω − (E∨) = K Ω − (E∨) ∗E −→ X ⊗ called the Hodge Star, where K = Ωn,0 = ∆nT (holomorphic line bundle) is called the canonical X C X∨ bundle of a complex manifold X.

1 ∗ q ∨ 0,q 0,q 1 Fact 5.1.3. ∂X = ( 1) − ∂KX E : (E) − (E) is the formal adjoint of ∂E. − ∗E ◦ ⊗ A −→ A

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

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

2 ∆ := dd∗ + d∗d = (d + d∗)

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 ∆ for ∆ = (∂ +∂∗ )2. E ∂E E E From construction,−→ 2 2 α, ∆ α = dα + d∗α h d i || || || || and analogously for the Hermitian case.

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

k kΩX , ∆d for the Riemannian case, (F, φ) = ⊕ 0,q qΩ (E), ∆ for the Hermitian case. ⊕ ∂E where φ : F F is an automorphism, and F = Ωk or F = Ω0,q(E). −→ ⊕k X ⊕q

65 5.2 Some applications of the Main Theorem

Theorem 5.2.1 (Riemannian case). Let k be the space of harmonic k-forms. Then the map H Hk Hk (X, R (or C)) given by α [α], which is well deﬁned since dα = 0, is an isomorphism. −→ dR 7→

k Proof: By the Main Theorem, β can be written as α + ∆γ = α + dd∗γ + d∗dγ, where α + ∈ A dd∗α is d-closed. Since β is closed we have d∗dγ is closed and hence it belongs to (Im(d∗))⊥. So 2 0 = (dγ, dd∗dγ) = d∗dγ . Similarly, we get the analogous theorem for ∆ where we can identify || || ∂E Hq(X, (E) = ) with H0,q(X,E) via the Dolbeaut isomorphism. O E ∂

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

Corollary 5.2.1. If X be a compact manifold then Hq(X, R) is ﬁnite dimensional. If X is a compact complex 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:

k k ΩX , ∆d if X is Riemannian, (F,P ) = Ω0,q(E), ∆ E X is a holomorphic vector bundle ( L q ∂E −→ where P is an elliptic or Laplacian-likeL operator. This implies that

C∞(F ) = Ker(P ) P (C∞(F )) ⊥ where Ker(P ) is ﬁnite dimensional.

Corollary 5.3.1.

k ∼ k k (1) There is an isomorphism HdR(X, Ror C) given by α [α], where is the ﬁnite dimensional space of harmonic forms. H −→ 7→ H

(2) There is an isomorphism 0,q ∼ Hq(X,E), where 0,q is the ﬁnite dimensional space of (0, q)-forms. H −→ H Both isomorphisms depend on the chosen metric.

Note that 1 k ∼= k HdR(X, R) H (X, R)

2 3 ∼= ∼=

Hˇ (H, R)

1 2 3 where ∼= is given by the de Rham isomorphism, ∼= and ∼= are given by the Poincar´eLemma. Recall the Dolbeaut isomorphism: p,q 0,q p ( ) p,q p p,q H (X,E) = H (X, Ω E) = ∗ H (X, Ω E) =: H (X,E) (ﬁnite dimensional), ∂ ∼ ∂ ∼ p p Ω E = (∆ T ∗ E) (space of holomorphic p-forms with values in E). ∼ O X ⊗ where ( ) comes from the isomorphism ΩpE = Ker∂0( p,0)(E) p,1(E)). Also, ∗ A −→ A p,q p,q q p H (X) = H (X, C) = H (X, ΩX ) (sheaf of holomorphic functions).

It is known that ∆ = ∂∆∂ in every K¨ahlermanifold. Then the following theorem follows:

Theorem 5.3.2. Let X be a K¨ahlermanifold. Then

k p,q H (X, C) = H (X). p+Mq=k k p,q p,q In other words, if [α] H (X, C) with α harmonic, then α = p+q=k α , where α is the (harmonic) component of type (p,∈ q). P

67 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 heat F (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:

inθ F (0, t) = an(t)e . X We have

2 inθ ∂ + ∆ = 0 = 0 = a0 (t) + n a (t) e t θ ⇒ n n n2t = a (t)X = a e− , where a = a (0), ⇒ n n n n n2t inθ = F (θ, t) = e− a e . ⇒ n n 0 X≥

It follows F (θ, t) a = f(θ)dθ = Av 1 (f), where the integral is the initial distribution. −→ 0 S1 S R Example 5.4.2. Let X = R. Doing the same exercise as above but using Fourier transforms, we get

1 (x−y)2 F (x, t) = e− 4t f(y)dy = e (x, y, t)f(y)dy. √4πt R ZR ZR 2 2 1 (x−y) 1 ||x−y|| The function e (x, y, t) = e 4t is called the heat kernel. Similarly, e n (x, y, t) = e 4t . R √4πt − R √4πt − 1 n2t in(x y) n2t inx iny n2t On S , we have e(x, y, t) = n e− e − = n e− e e , where e− are the eigenvalues of ∆, and einx and einy are the eigenfunctions. In general, the existence of e (x, y, t) is diﬃcult to obtain analytically P P X but 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

(∂ + ∆)A(t) = 0 ( ) t ∗ A(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 ( ). || || ∗

2 2 2 Proof: ∂ A(t) = 2 ∂ A, A = 2 ∆A, A = 2 dA + d∗A 0. t|| || h t i − h i − || || || || ≤

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

A(t) = K ( , y, t)α(y)dVol(y) p · ZX solves ( ), for every α p(X). ∗ ∈ A

Let T (x) = K( , y, t)α(y)dy. t X · R

Theorem 5.4.2. Tt satisﬁes:

(1) Tt1+t2 = Tt1 Tt2 .

(2) Tt is formally self-adjoint.

(3) T α tends to a C∞ harmonic form H(α) as t . t −→ ∞ (4) G(α) = ∞(T α Hα)dt is well deﬁned and yields the Green operator G, i.e. 0 t − R 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 = ∂tTtη, Tτ = ∆Ttη, Tτ = Ttη, ∆Tτ = Ttη, ∂tTτ = ∂t Ttη, Tτ . This impliesh thati , h is a functioni − of ht + τ, so denotei − h , by g(it + τh). Therefore,i h i h i h i T η, = g(t + 0) = g(0 + t) = η, T . h t i h τ i (3) (1) + (2) = h > 0, ⇒ ∀ T α T α 2 = T α 2 + T α 2 2 T α, T α || t+2h − t || || t+2h || || t || − h t+2h t i = T α 2 T α 2 2( T α 2 T α T α ) || t+2h || − || t || − || t+h || − || t+2h |||| t || 2 2 and Tαα converges, and therefore it is decreasing. Hence Tt+2hα Ttα 0. It follows || || L2 || − || −→ T α H(α), for some H(α) p(X) , called the harmonic projection. Fix τ > 0, then t −→ ∈ A Ttα = Tτ Tt τ α H(α) := Tτ H(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: − R ∞ ∞ ∆G(α) = ∆T αdt = ∂ T αdt = α H(α), t − t t − Z0 Z0 and for β harmonic we have

∞ ∞ Gα, β = (T H)α, β dt = α, (T H)β dt = 0, since (T H)β = 0. h i h t − i h t − i t − Z0 Z0

69 Corollary 5.4.1. There exists an orthogonal direct sum decomposition

p p p 1 p p 1 (X) = (X) d( − (X)) = (X) d∗( − (X)) A H A H A M M where Im(∆) = Im(d) + Im(d∗).

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

p Then (X) = = α + dd∗γ + d∗dγ, where α (X). Also, O ∈ A ⇒ O ∈ H 2 d∗dγ = d∗dγ, α = dγ, dα , where dα = 0. || || h i h i

70 5.5 Index Theorem (Heat Equation approach)

p p p p p Let ∆ : (X) (X), λ R 0. Let E denote the λ-eigenspace for ∆ (ﬁnite dimensonal). The A −→ A ∈ ≥ λ square root √∆ = δ is called the Dirac operator.

Lemma 5.5.1. The sequence 0 E0 d E1 d En 0 −→ λ −→ λ −→ · · · −→ λ −→ is exact for λ > 0.

Proof: ω Ep = ∆p+1dω = d∆pω = λdω = dω Ep+1. ∈ λ ⇒ ⇒ ∈ λ p p 1 Now ω E and dω = 0 = λω = ∆ ω = d∗d + dd∗ω = ω = d d∗ω . Then ∆d∗ω = d∗∆ω = λd∗ω. ∈ λ ⇒ ⇒ λ

Corollary 5.5.1. ( 1)pdim(Ep) = 0. p − λ P

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

p t∆p p tλ(p) p 0 λ(p)(t) ( 1) tre− = ( 1) e− i = ( 1) e− i − − − p p p i X X X X (p) where i0 is over i where λi = 0. P

(p) λi (t) p Note that i0 e− = dim(Ker(∆ )). Hence

P p p p t∆(p) t∆(p) (X) = ( 1) dim(Ker(∆ )) = ( 1) tre− , where e− = T , X − − t p p X X = ( 1)p e(p)(x, x, t)dVol(x). − p i X X X Z

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

Hence, as t 0, we have −→ 1 n/2 ∞ ∞ ( 1)ptrup(x, x)dVol(x) tk. X ∼ 4πt − k X p=0 ! kX=0 Z X

71 This implies ∞ p 0 if k = n/2, 4πn/2 ( 1)tru (x, x)dVol(x) = 6 − k (X) k = n/2. X p=0 Z X X

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

(X) = ω, X ZX where ω is given in a local frame by

ω = cn (signσ)(signτ)Rσ(1)σ(2)τ(1)τ(2) Rσ(n 1)σ(n)τ(n 1)τ(n) ··· − − σ,τ X ( 1)n/2 and cn = −n/2 n . (8π) ( 2 )!

For n = 2, ω = 1 (R R R R )dA = 1 RdA = K dA where K is the Gaussian curvature. 8π 1212 − 1221 − 2112 − 2121 − 2π 1212 2π

72 BIBLIOGRAPHY

[1] Griﬃths Phillip; Harris Joseph. Principles of Algebraic Geometry. Pure & Applied Mathematics. John Wiley 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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