COMPLEX GEOMETRY Course notes

MARCO A. PÉREZ B.
Université du Québec à Montréal.
Département de Mathématiques.
COMPLEX GEOMETRY
Course notes
December, 2011.

These notes are based on a course given by Steven Lu in Fall 2011 at UQÀM. 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

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 ⊕ i Hi dR
(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 Kähler 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
iv

Chapter 1
COMPLEX ANALYSIS
1.1 Complex Analysis in one variable
Let U ⊆ C = R 2 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 ′ (z 0 ) = df
dz (z 0) = lim z→z0
f(z)−f(z 0)
z−z 0
exists for every z 0 ∈ U. We shall denote this by f ∈ O(U). If S ⊆ C is any subset, we shall say that f is
holomorphic 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 ∂f
∂x
dx +
∂y makes sense and df(x, y) ∈ Hom R(T z=x+iy U, R 2 ).
Recall that
dz = dx + idy and dz = dx − idy
Using these expressions, we can write the differential df as
df = 1 2
( ) ( )
∂f
∂x − i ∂f
∂y
dz + 1 ∂f
2 ∂x + i ∂f
∂y
dz = ∂f ∂f
∂z
dz +
∂z dz
Notice the following relations
( )
∂f
∂z
= ∂f
∂z
and
( )
∂f
∂z
= ∂f
∂z
Recall that
df =
f is complex differentiable
⇐⇒
f is R-differentiable and ∂f
∂z
= 0 (Cauchy-Riemann condition)
⇐⇒
∂u ∂v
∂z
= −i
∂z
( )
⇐⇒
ux u y
is a rotation matrix up to a real scalar multiple (u
v x v x = v y and u y = −v x ).
y
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 ∈ O(U) if and only if ∫ f = 0, for every V ⊂⊂ U simple connected.
∂V
Theorem 1.1.2 (Cauchy’s Integral Formula). z 0 ∈ V ⊂⊂ U if and only if
f(z 0 ) = 1 ∫
2πi ∂V
f(z)
z − z 0
dz.
If V = D ɛ (z 0 ) is a disk centered at z 0 of radius ɛ, then we shall denote the previous integral by
f(z 0 ) = Avg ∂V (f) := 1
2π
∫ 2π
0
f(z 0 + ɛe iθ )dθ.
Corollary 1.1.1 (Liouville Theorem). Every holomorphic function on C is constant.
Proof: Let V ɛ = B ɛ (z 0 ), z 0 ∈ C. We show that f ′ (z 0 ) = 0. Using the Cauchy’s Integral formula, we
have
f ′ (z 0 ) = 1 ∫
f(z)
2πi (z − z 0 ) 2
Notice that |f| is bounded on ∂V ɛ . Then |f(z)| ≤ M on ∂V ɛ for some M > 0. So we have
|f ′ (z 0 )| = 1
∫
∣ f(z) ∣∣∣
2π ∣
∂V ɛ
(z − z 0 ) 2 dz ≤ 1 ∫
|f(z)|
2πɛ 2 ∂V ɛ
|z − z 0 | 2 dz
= 1 ∫
2πɛ 2 |f(z)|dz ≤
M ∫
∂V ɛ
2πɛ 2 dz
∂V ɛ
= M
2πɛ 2 · 2πɛ = M ɛ
It follows |f ′ (z 0 )| −→ 0 as ɛ −→ ∞. Hence f ′ (z 0 ) = 0 for every z 0 ∈ C and f is constant in C.
∂V ɛ
Corollary 1.1.2 (Riemann Extension Theorem). If f ∈ O(U − z 0 ), bounded near z 0 and continuous at z 0 ,
then f ∈ O(U).
Proof: If f ∈ O(U − z 0 ) then f ∈ O(U − B ɛ (z 0 )), for some ɛ > 0. Then the result follows since the
Cauchy’s Integral Formula still holds in this case.
2

Theorem 1.1.3 (Local Structure of f ∈ O(U)). If f ∈ O(U) is non-constant at z 0 ∈ U. Let
m = min{n > 0 / f (n) (z 0 ) ≠ 0}.
Then there exists a bi-holomorphic function ϕ : V −→ W from a neighbourhood of V of z 0 to a neighbourhood
W of 0 ∈ C with ϕ(z 0 ) = 0 such that
f(z) − f(z 0 ) = ϕ(z) m , for every z ∈ V.
Proof: Note that f(z) − f(z 0 ) = (z − z 0 ) m g(z) with g(z 0 ) ≠ 0 and g ∈ O(U). Since the quotient
f(z)−f(z 0)
z−z 0
is bounded on U − z 0 and continuous at z 0 , we have by the Riemann Extension Theorem that
(z − z 0 ) m−1 g(z) = f(z)−f(z0)
z−z 0
is holomorphic on U. Proceeding this way, we have that g(z) ∈ O(U).
We study several cases: If n = 1 then f ′ (z 0 ) ≠ 0 and by the Inverse Function Theorem we can choose
ϕ(z) = f(z) − f(z 0 ). Now assume n ≠ 1. Since g(z 0 ) ≠ 0 then g(z) ≠ 0 on a neighbourhood of z 0 . So
we can write g = h m on a neighbourhood V . We have
f(z) − f(z 0 ) = [h(z)(z − z 0 )] m
with ϕ ′ (z 0 ) = h(z 0 ) ≠ 0. Hence, up to a local change of coordinates, f is locally of the form z ↦→ z m for
some m. Such a number m is called the ramification degree of f at z 0 .
Corollary 1.1.3 (Open Mapping Theorem). If f ∈ O(U) is non-constant and U is connected, then f is an
open mapping.
Corollary 1.1.4. If f ∈ O(U) and |f| has a local maximum at z 0 ∈ 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 B ɛ (f(z 0 )) ⊆ f(U)
for some ɛ > 0. In this neighbourhood there are some points of modulus greater that 0. Hence f(z 0 ) 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 z 0 = (x 0 , y 0 ) ∈ U there exists a
neighbourhood V of z 0 such that
f(z) = ∑ ∞
α,β=0 a α,β(x − x 0 ) α (y − y 0 ) β for every z ∈ V.
Similarly, f is said to be complex analytic on U if for every z 0 ∈ U there exists a neighbourhood V of z 0
such that
f(z) = ∑ ∞
n=0 a n(z − z 0 ) 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
On the other hand,
f(z) = 1 ∫
f(w)
2πi ∂D r(z 0) w − z dw.
(
1
w − z = 1 1
w − z 1 − z−z0
w−w 0
)
1
=
w − z 0
∞ ∑
n=0
( z − z0
w − z 0
) n
.
It follows that
where
and |w − z 0 | = r on ∂D r (z 0 ).
f(z) = 1 ∫
2πi ∂D r(z 0)
f(z)
∞∑ (z − z 0 ) n
∞
(w − z 0 ) n+1 dw = ∑
a n (z − z 0 ) n
n=0
n=0
a n = 1 ∫
f(w)
2πi ∂D r(z 0) (w − z 0 ) n+1 dw = 1 n! f (n) (z 0 )
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 Taylor
expansion of f about γ,
∞∑ f (n)
f(z) = (z − γ) n
n!
n=0
for every z ∈ D r (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) ∩ D r (γ). We have
0 =
∞∑
n=0
f (n)
(z − γ) n
n!
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 ∈ 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 ∂D r for every r ∈ (−ɛ, 1), then
f(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, D
denotes the Poncaré disk, 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 ⊆ C n be an open subset.
Definition 1.3.1. A function f : U −→ C in C R ′ (U) (R-differentiable on U) is called holomorphic, denoted
f ∈ O(U), if for every u ∈ U, the differential df u ∈ Hom(T u U, 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 z 0 ∈ U, f has the form
f(z 0 + δ) = ∑ I
a I δ I (normal convergence)
where I = (i 1 , . . . , i n ) and z I = z i1
1 · · · zin n .
(3) If D = {(ζ 1 , . . . , ζ n ) / |ζ i − a i | < α i ∈ R >0 } is a poly-disk in U, then for every z = (z 1 , . . . , z n ) ∈ Int(D)
( ) n ∫ 1
dζ 1
∧ · · · ∧
dζ n
2πi
ζ 1 − z 1 ζ n − z n
where δD = {|ζ i − a i |} = α i ⊂ ∂D.
|ζ i−a i|=α i
f(ζ)
Note that if f ′ ∈ C R ′ (U) then
df = ∑ ∂f
∂x i
dx i + ∑ ∂f
∂y i
dy i = 1 ∑ ( )
∂f
2 ∂x i
− i ∂f
∂y i
dz i + 1 ∑ ( )
∂f
2 ∂x i
+ i ∂f
∂y i
dz i
Theorem 1.3.2. Let f ∈ C n+1 such that f(0) = 0. Write C n+1 = {(w, z 1 , . . . , z n ) = (w, z)}. If f ≢ 0 on
the w − axis (z = 0) then on some neighbourhood V of 0, we have
where g is never zero on V .
f = (w, z)(w d + a 1 (z)w d−1 + · · · + a d (z))
Denote p(w, z) = w d + a 1 (z)w d−1 + · · · + a d (z). Hence locally we have zero(f) = zero(p). It follows that
the roots of p are single valued holomorphic functions w = b i (z) away from the discriminant locus of f
(∆ f (z) = 0, where ∆ f (z) is a polynomial in the a i ’s). Hence {f = 0} is a étale cover that covers the hyperplane
{w = 0} = {(z 1 , . . . , z n )}. 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.
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) = {(z 1 , z 2 ) / |z 1 | < r and |z 2 | < 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 ⊆ C n has complex codimension greater or equal than 2, where S is an analytic subset
and f ∈ O(U − S), then f extends to a holomorphic function on U.
Proof: We only proof the second part.
(1) Take a slice z 1 = const. Then U − V = {r ′ < |z 2 | < r} on this slice. Set
F (z 1 , z 2 ) = 1 ∫
f(z 1 , w 2 )
dw 2
2πi |w 2|=r w 2 − z 2
Hence F : U −→ C is holomorphic in z 1 since
∂f
= 0 =⇒ ∂F
∂z 1 ∂z 1
and clearly also in z 2 (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 ∈ 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.
7

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 C n , 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:
(1) CP n = {[z 0 : · · · : z n ]} = C n+1 − {0}/ ∼, where
(z 0 , . . . , z n ) ∼ (z ′ 0, . . . , z ′ n) ⇐⇒ z = tz ′ , for some t ∈ C ∗ .
Let U 0 = {[z 0 : · · · : z n ] / z 0 ≠ 0}. Let ϕ 0 : U 0 −→ C n be the map given by
(
z1
[z 0 : · · · : z n ] ↦→ , . . . , z )
n
z 0 z 0
which we shall call the 0-th affine chart. Note that there exist n + 1 affine charts that cover CP n .
(2) The compact complex torus C n /Γ, where Γ ∼ = Z n .
(3)
PGL(1, 0) = Aut(CP 1 ) = set of Moebius transformations
{ }
az0 + bz 1
=
/ ad − bc ≠ 0 /{±1}
cz 0 + dz 1
The following sets are Riemann surfaces:
(4) CP 1 .
9

(5) C and C ∗ = C − {0}.
(6) The half-plane model H = {z / Im(z) > 0}.
(7) The Poincaré disk model ∆ = D = {z / |z| < 1} and ∆ ∗ = ∆ − {0}. The models H and D are related
by the map
z ↦→ z − a
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
elliptic curve. The manifold CP 1 is called a rational curve.
Note that CP 1 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(z 0 , . . . , z n ) be a homogeneous polynomial. Then
C = V (f) = {[z 0 : z 1 : z 2 ] / f(z 0 , z 1 , z 2 ) = 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).
(2) x d + y d + z d gives the Fermat curve of degree d in CP 2 . It is smooth since df ≠ (0, 0, 0) on C,
df = (x d−1 dx, y n−1 dy, z n−1 dz).
10

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) 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 f 1 , f 2 : S 1 −→ S 2 are mappings of Riemann surfaces such that they
coincide on a non-discrete subset of S 1 , then f 1 ≡ f 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 : S 1 −→ S 2 be a non-constant map of Riemann surfaces. Assume that S 1 is
compact. Then f is surjective and S 2 is compact. Furthermore, if f is proper (i.e., f −1 (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 ↦→ z d , 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
mult p (f) = d.
The degree of f is defined by
deg(f) = ∑ p∈F mult p(f)
for every fibre F . The ramification index of f at p is
r p (f) = mult p (f) − 1.
A map is said to be unramified if r p (f) = 0 for every p ∈ S.
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) lim z→p f(z) = ∞.
(b) f can be written locally as a Laurent series
∞∑
f(z) = a i z i
−∞
with a i = 0 for every i < n ∈ Z.
(c) f = g/h, where g, h ∈ O(p), g(p) ≠ 0 and h(p) = 0.
The set of such functions is denoted M(Z). We have
f ∈ M(Z) ⇐⇒ f : Z hol
−→ CP 1
and that
poles of f = f −1 (∞)
Example 2.3.1.
(1) A non-constant polynomial defines a meromorphic function from CP 1 with pole order at ∞ equal to
deg(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 : Z 1 −→ Z 2 corresponds to a finite field extension
f ∗ : M(Z 2 ) ↩→ M(Z 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 Res p (ω).
Theorem 2.3.1 (Residue). V ⊂⊂ Z with rectifiable boundary ∂V and ω differentiable on Z. Then
∫
ω = ∑ Res p (ω)
p∈V
∂V
12

hol
We shall denote the space of meromorphic differentials by M ′ (Z).
Theorem 2.3.2. ω ∈ M ′ (Z), ∑ p∈Z Res p(w) = 0.
Corollary 2.3.1 (Sum rule or product rule, Reciprocity Theorem). If Z is compact and f ∈ M(Z), then
on Z counting multiplicity.
#zero(f) = #poles(f)
Corollary 2.3.2. If f ∈ M(CP 1 ) then f is rational. Hence M(CP 1 ) = 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 a
meromorphic differential (ω ∈ M ′ (S)). Then we have
∫
ω = ∑ Res p ω
∂U
p∈U
If ∂U = ∅ (so U = S is compact) then
∑
Res p ω = 0
p∈S
We have that if S is compact then S = R(S), the set of rational functions (later, we are going to study this
in detail).
Example 2.3.2. M(P 1 ) = R(P 1 ), where P 1 = CP 1 . Note that CP 1 is compact and CP 1 = C ∪ C, where
there is a map between charts
z ∈ C w= 1 z
↦→ w ∈ C
Since ∑ p∈S Res pω = 0, we have ω = df f , and Res pω = ord p f.
Recall that f ∈ M(P 1 ) if and only if f : P 1 −→ P 1 is holomorphic.
Example 2.3.3. Let C be a quadratic (conic) curve in CP 2 = P 2 . Let p ∈ C. The space of lines passing
through p is P 2 and so gives a meromorphic map P 2 −→ P 1 . We have the following diagram:
P 2 P 1
P 2 − {p}
hol
C − {p}
13

This shows that C ∼ = hol P 1 . For example, we have the Fermat curve z 2 0 + z 2 1 = z 2 2. If P 1 = {[u : v]}, then
define a map z 0 = (u − v) 2 , z 1 = 2uv and z 2 = (u + v) 2 .
Note that π 1 (S) = 0 where S = P 1 , C, D. Recall that if π 1 (S) = Z then S is not compact.
Example 2.3.4. S 1 = C ∗ , S 2 = D − 1 2D. These two examples are not biholomorphic Riemann surfaces.
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
ω ↦→ w − a
w − a
14

2.4 Weierstrass P -function on C
Consider the torus C/Γ, where
Γ = 〈1, τ〉 τ∈C
= Z ⊕ Zτ
is a lattice in C. Define the Weirstrass P -function by the formula:
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.
P(z) = p(z, Γ) = 1
z
+ ∑ 2 λ∈Γ−{0}
[
]
1
(z−λ)
− 1 2 λ 2
Hence P descends to a meromorphic function on C = C/Γ with a pole at 0 (degree = 2), i.e., we have a
holomorphic function
C −→ f
CP 1
Locally, this map looks like z ↦→ z n about 0. In this case n = 2. This map has exactly degree 2 since there
are no other poles. Moreover, f is branched at 0.
0
∞
CP 1
Now P ′ (z) is also periodic (period Γ) with triple poles on Γ and no other poles. The map C −→ P 2 given by
defines a holomorphic function C − {0} −→ P 2 ,
and it extends over 0.
z ∈ C ↦→ [P(z) : P ′ (z) : 1]
[P(z) : P ′ (z) : 1] =
15
[ ]
P(z)
P ′ (z) : 1 : 1
P ′ (z)

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 = ∑ a i P i
where a i ≠ 0 for every i and {P i } is a discrete subset of S. A divisor D is called effective if a i ≥ 0 for every i.
If supp(D) = {P i / a i ≠ 0} is finite, then
deg(D) := ∑ a i
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 = ∑ (ord P f)P and (f) ∞ = ∑ P ∈f −1 (∞) (mult P f)P
Note that mult P f = −ord P f, so we can rewrite the previos expresion as
(f) = ∑ (ord P 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 a
map 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 D 1 and D 2 are said to be linearly equivalent, denoted D 1 ∼ D 2 , if D 1 −D 2
is principal.
Example 2.5.1. D 1 ∼ D 2 on P 1 if and only if deg(D 1 ) = deg(D 2 ).
Example 2.5.2. What condition we need if we want D 1 ∼ D 2 on C = C/Γ. Let p, q ∈ C be two distinct
points in C and suppose that D = p − q = (f) for some f ∈ M ∗ . Then f is a map C −→ P 1 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. ord p ω = ord p f. The divisor of the form
(ω) := ∑ p∈S (ord pω)P
is called a canonical divisor and is denoted K S or simply K. A divisor is canonical if D ∼ (w).
16

Example 2.5.3.
• K C/Γ = (dz) = 0 · P .
• K P 1
= (dz) = −2 · ∞, z = 1 dω
ω
, dz = −
ω
. 2
Theorem 2.5.1 (Riemann - Hurewicz). If f : S 1 −→ S 2 is a finite map, then K S1
is the ramification divisor
R = ∑ r p (f)P,
p∈S 1
∼ K + S 2 + R where R
where each r p (f) ≥ 0.
Proof: Locally, f looks like z ↦→ z n , and dz n = nz n−1 dz.
Corollary 2.5.1 (Riemann - Hurewicz formula).
2 · g(S 1 ) − 2 = degK S1 = (degf) · degK S2 + degR,
where K S1 = (ω), ω ∈ Γ(T ∨ S 1 ) (global sections of the tangent bundle) ←→ ω ∨ ∈ Γ(T S 1 ).
17

2.6 Covering spaces
Recall that an étalé space (or étale 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 image
p(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 from
the following important property: if the map p : ˜X −→ X is a universal cover of the space X and the map
p ′ : X ′ −→ X is any cover of the space X where the covering space X ′ is connected, then there exists a
covering map f : X ′ −→ ˜X such that p ◦ f = p ′ . Any manifold X has a universal cover ˜X with étale covering
map f : ˜X −→ X and π1 (X) acts on ˜X discretely and freely with quotient f. Moreover, there exists a
bijection
étale
π 1 (S) ⊇ H ↦→ { ˜X/H −→ X}
between the set of subgroups of π 1 (X) up to conjugation and the set of étale coverings from a connected
manifold.
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 ∗ π 1 ( ˜X, ˜x) is normal in π 1 (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 : Z 1 −→ Z be a finite étale covering corresponding to H ↩→ π 1 (Z). Then there exists
a finite regular covering h : Z 2 −→ Z and a finite étale covering g : Z 2 −→ Z 1 such that f ◦ g = h.
Z 1 Z 2
étale covering
Z
∃regular
Proof: H has a finite number of conjugates in π 1 (Z), such a number equals [π 1 (Z) : N(H)] and these
intersection is then of finite index.
Corollary 2.6.1. For every n, there exists a unique étale n-sheeted covering Z −→ D ∗ and it is isomorphic
to D ∗ −→ D ∗ (z ↦→ z n ).
Example 2.6.1. If f : Z 1 −→ Z 2 is finite between Riemann surfaces, then there is a finite étale covering
where ∆ = f(suppR f ) is the branching locus.
Z 1 − f −1 (∆) −→ Z 2 − ∆
18

Conversely, the previous corollary implies:
Theorem 2.6.1. Let ∆ ⊆ Z 2 be a discrete subset. A finite étale covering U −→ Z 2 − ∆, where U is an open
subset, has a unique continuation to a finite map
where Z 1 is a Riemann surface.
U ⊆ Z 1 −→ Z 2
19

2.7 The Riemann surface of an algebraic function
Let Z 2 be a Riemann surface.
Proposition 2.7.1. Let
P (T ) = T n + c 1 T n−1 + · · · + c n
in M(Z 2 )[T ] be an irreducible polynomial. Then there exists a map of Riemann surfaces f : Z 1 −→ Z 2 of
degree n and a meromorphic function F ∈ M(Z 1 ) that satisfies:
F n + f ∗ (c 1 )F n−1 + · · · + f ∗ (c n ) = 0.
(∗)
Proof: Let ∆ ⊆ Z 2 be the discrete set containing the poles of the c i ’s and the points p where
P p (T ) := T n + c 1 (p)T n−1 + · · · + c n (p)
has multiple roots. Then U = {(p, z) ∈ (Z 2 − ∆) × C / P p (z) = 0} is a Riemann surface, and Z 2 − ∆ is
a finite étale cover. We claim that it is connected, i.e.,
Claim: Given f : Z 1 −→ Z 2 , then every F ∈ M(Z 1 ) is algebraic over M(Z 2 ) and satisfies an equality
of the form (∗) but with degree less or equal than the degf.
Corollary 2.7.1. If Z 1 −→ Z 2 is finite, then
is a finite field extension.
f ∗ : M(Z 2 ) −→ M(Z 1 )
Example 2.7.1. M(P 1 ) = C(Z), the field of rational functions in variable z.
Theorem 2.7.1. As soon as there exists a meromorphic function f on Z 1 (←− compact =⇒ f is finite),
M(Z 1 ) is finite algebraic over C(z) of extension deg = degf.
Z 1
f
−→ P 1
Corollary 2.7.2.
(1) If Z 1
f
−→ Z 2 is finite then M(Z 2 ) f ∗
↩→ M(Z 1 ) is a finite field extension of degree (f).
(2) Conversely, let M(Z 2 ) ϕ
↩→ M(Z 1 ) = K be a finite field extension of degree d. Then there exists a finite
map Z 1 −→ Z 2 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 Riemann
surface. 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:
( )
ω 1 , ω 2 ∈ M ′ ω1
(Z) =⇒ = (f) ∈ Div P (Z), where f ∈ M(Z).
ω 2
Here, Div P denotes the set of principal divisors. Note that ω ∈ M ′ (Z) if and only if ω ∈ Γ m (T ∨ Z), where
T ∨ Z denotes the cotangent bundle of Z and Γ m (T ∨ Z) is the set of holomorphic sections of T ∨ Z. Also,
T ⊗ T ∨ = O (trivial line bundle) and so
( ) 1
1 = · ω ∈ Γ m (O) = O,
ω
where 1 ω ∈ T Z and ω ∈ T ∨ Z.
If ω 1 = fω 2 then (ω 1 ) = (f) + (ω 2 ). So we get the formula
deg( ) = deg(f) + deg( )
On the other hand, deg(f) = 0. To show this, we know that f(z) = z n locally, where n = ord(f). Then
df
f
= n dz
z and ∑ p∈Z Res p df f
= 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 and
connected Riemann surface.
Let g be the topological genus of Z. We have the following relations:
deg(ω) = 2g − 2 and deg ( 1
ω
)
= 2 − 2g = χ(Z)
where 1 ω
∈ Γ(T Z).
genus
= # of holes
Recall the following results:
21

Theorem 2.8.2. Let P (T ) = T n + c 1 T n−1 + · · · + c n ∈ 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
F n + (f ∗ c 1 )F n−1 + · · · + (f ∗ c n ) = 0
Corollary 2.8.1.
(1) Z 1 −→ Z 2 finite =⇒ f ∗ : M(Z 2 ) ϕ
↩→ M(Z 1 ) is a finite field extension of degree n.
(2) Conversely, any finite field extension of degree n gives rise to a finite map Z 1 −→ Z 2 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
to M(Z) for some Riemann surface Z (finite over CP 1 ).
Z is called a smooth model of K.
Theorem 2.8.3. Let C ⊆ CP 2 be an irreducible plane curve, i.e., C = V (F ) where F is an irreducible
homogeneous polynomial in (z 0 : z 1 : z 2 ). Then there exists a compact Riemann surface Z and a generically
injective map f : Z −→ CP 2 whose image is C.
Here, generically injective means birational (←→ isomorphic on an open set).
Proof: If F is irreducible then C[x 0 , x 1 , x 2 ]/(F, x 2 − 1) is an integral domain and has a quotient field
K. The mapping f(P ) = (x 0 (P ) : x 1 (P ) : 1) extends to by continuity to a desingularization of C.
Theorem 2.8.4 (Hurwitz). If f : Z 1 −→ Z 2 is a finite map, then K Z1 ∼ f ∗ K Z2 + R (f ∗ = pullback) where
R is the ramification divisor ∑ p∈Z 1
r p (f) · p, r p (f) = ord p (f) − 1, where f : Z 1 −→ Z 2 is locally at p of the
form f(z) = z ordp(f) and
deg(f) =
∑
ord p (f),
p∈ fibre
where this sum is independent of the choice of the fibre.
Corollary 2.8.2 (Riemann-Hurwitz).
deg(K Z1 ) = (deg(f)) · (deg(K Z2 )) + deg(R)
Corollary 2.8.3. g(Z 1 ) ≥ g(Z 2 ), 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(K 2 )) = dim C (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 manifold
is orientable if its transition functions have positive Jacobian . Let Z be a Riemann surface, consider two
intersecting charts U α and U β , and let z ∈ U α ∩ U β . Then we have
df
dz (z) = f ′ (z)
as an R-matrix.
We write f = u + iv, then df = α + iβ.
β
α ◦ β −1
holomorphic
α
C
By the Cauchy-Riemann equations, we have
( α β
df =
−β α
)
and so
Jac(α ◦ β −1 ) = det(D(α ◦ β −1 )) = |df| 2 = det(df) = α 2 + β 2 > 0.
It is also known that every Riemann surface is obtained by attaching handles to CP 1 = S 2 .
# 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 attaching
handles to S 2 = CP 1 = C ∪ {∞}. We denote Z g for a Riemann surface Z of genus g. It is known that the
first homotopy group of Z g is given by
π 1 (Z g ) = Z a1 ∗ Z b1 ∗ · · · ∗ Z ag ∗ Z bg / 〈 a 1 b 1 a −1
1 b−1 1 · · · a g b g a −1
g
Notice that every hole gives two generators.
b −1
g
〉
b 1
b g
a g
a 1
a 1
b 1
a −1
1
b −1
1
b −1
g
Let A i denote the space of C ∞ -complex valued i-forms on a Riemann surface, then
A = A 1,0 ⊕ A 0,1 ,
ω ′ = fdx + gdy = hdz + hdz,
where hdz ∈ A 1,0 and hdz ∈ A 0,1 . We also have differential operators making the following diagram commutes
d
A 1
d
C ∞ − functions = A 0 ∂
A 0,1 ∂
A 2
∂
A 1,0
∂
where d = ∂ + ∂.
Definition 2.9.1. An i-form ω ∈ A i 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 the
quotient
HdR i (Z) = {closed i-forms}/{exact i-forms}
25

Note that HdR 0 (Z) = C if Z is connected. If Z is also compact, by the Poincaré duality Theorem we have an
isomorphism HdR 2 (Z) −→ H0 dR (Z) given by ∫
[ω] ↦→
Also, HdR n (Z) = 0 for every n ≥ 3. For
cohomology is given by
Z
ω
any compact and connected Riemann surface Z g, the middle
H 1 dR (Z g) = π 1 (Z g )/ 〈commutator subgroup〉 = Z a1 ⊕ Z b1 ⊕ · · · ⊕ Z ag ⊕ Z bg
Theorem 2.9.2. Let ω be a holomorphic differential (←→ 1-form). Then ω is d-closed (hence [ω] ∈ H 1 dR (Z)).
Proof: Locally, ω = f(z)dz where f ∈ O. Then
dω = (∂ + ∂)ω = ∂ω + ∂ω =
( ) ( )
∂f
∂f
∂z dz ∧ dz +
∂z dz ∧ dz
= ∂f ∧ dz + ∂f ∧ d = 0 + 0dz ∧ dz, since f is holomorphic.
26

2.10 Product structures on ⊕ i Hi dR (Z)
By taking wedge products of forms, we have a surjective map HdR 1 (Z) × H1 dR
(Z) −→ C given by
∫
([ω 1 ], [ω 2 ]) ↦→ ω 1 ∧ ω 2
In this situation we have H 1 dR (Z) ∼ = (H 1 dR (Z))∨ (finite dimesnion) and so the previous map is a perfect pairing.
Z
We also have the cap product ∩ : H 1 (Z) × H 1 (Z) −→ Z which gives rive to a perfect pairing.
By the Poincaré duality Theorem we have HdR 1 (Z) ∼ = H 1 (Z) ∨ ⊗ C. Also
HdR(Z) 1 = (H 1 (Z) ∗ = H 1 (Z)) ⊗ C.
Recall that the Euler characteristic of Z is given by
χ(Z) := ∑ i (−1)i dimHdR i (Z) = ∑ i (−1)i dim Z H i (Z) = 2 − 2g
Recall H 0 dR (Z) = H2 dR (Z) = C and H1 dR (Z) = C2g .
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
P g (Z) = dim C (Ω)
Clearly, Ω ↩→ HdR 1 (Z) if Z is compact. We have that
fdz = dg dz = dg = 0
dz
implies that g ∈ O. Since Z is compact, we get that g = const. Similarly, Ω ↩→ H 1 dR (Z). It follows 2P g ≥ 2g.
It is difficult to determine when they are equal. The fact that P g ≥ 2 implies that there exist meromorphic
functions. If P g = 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
is called the period map.
ω ↦→ ∫ C ω
Definition 2.10.2. Let S be a Riemann surface, define
H 1 (S; Z) = π 1 (S)/[π 1 (S), π 1 (S)]
where [π 1 (S), π 1 (S)] denotes the commutator subgroup of π 1 (S).
Note that if S is compact then H 1 (−) is a free Z-module generated by a 1 , b 1 , . . . , a n , b n . We can consider
each a i and b i as maps S 1 −→ S or curves with parameter t ∈ [0, 1] in S, oriented counterclockwise.
27

1
b g
a g
a 1
Since S is oriented we have that there exists an intersection pairing in H 1 (any two loops, or elements in H 1 ,
are homotopic to loops which are transversal). Note that a and b are the same element in H 1 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
C
p
−→ a
x
We have −→ a ∧ p
−→ b = kx ∧p y, and denote
and
(a, b) p = sign(k)
(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
( , ) : H 1 × H 1 −→ Z
If S is a Riemann surface of genus g, we have in terms of a basis that
(a i , b j ) = (b i , b j ) = 0, for every i, j,
(a i , b j ) = −(b j , a i ) = δ ij .
28

Then we have that each (a i , b j ) is a skew-linear pairing which is unimodular (det = 1).
a i
b i
a i 0 1
b i −1 0
det
1
Hence H 1 (S; Z) = Hom(H 1 (S; Z), Z) =: H 1 (S; Z).
Let ω ∈ M ′ (S) be a closed meromorphic differential. Recall that ω is exact if and only if ω = df for some
meromorphic function f ∈ M(S). Can we choose a function f ∈ M(S) such that f = ∫ z
ω? As an exercise,
p
think if it is possible that Res p (ω) = 0 for every p ∈ S. Consider the period homomorphism
Π ω : H 1 (S; Z) −→ C
∫
[γ] ↦→ ω, γ is a loop.
γ
This map is well defined. For if γ = γ ′ in H 1 then γ − γ ′ = ∂u, and by the Stokes Theorem we have
∫
γ ω = ∫ γ ′ ω. Also, Π ω is a C-linear map.
By the de Rham Theorem, we have an isomorphism HdR 1 (Z) ∼ = H 1 (S; Z) ⊗ C if Z is compact.
isomorphism is given by
[ω] ∈ HdR(Z) 1 ↦→ Π ω
Such an
Corollary 2.10.1. dim C HdR 1 (S) = 2g.
Proof: We have
2 − 2g = χ(S)
= ∑ (−1) i h i dR(S), where h i dR (S) = dim(H1 dR (S)),
= h 0 dR(S) − h 1 dR(S) + h 2 dR(S)
= 1 − dim C HdR(S) 1 + 1.
Theorem 2.10.1. The de Rham ismomorphism carries wedge product of forms defined by
isomorphically to the (cup) product on H 1 (S).
HdR 1 × HdR 1 −→ C
∫
([ω 1 ], [ω 2 ]) ↦→ ω 1 ∧ ω 2
S
29

Recall that Ω denotes the space of holomorphic 1-forms on S. Since
H 1 dR
∼ =
−→ H 1 (S; Z) ⊗ C = Hom C (H 1 , C)
if S is compact, we have an inclusion Ω ↩→ HdR 1 if S is compact. We have
ω exact =⇒ ω = df =⇒ f holomorphic =⇒ f constant =⇒ ω = 0
Definition 2.10.3. P g = dim(Ω) (geometric genus).
It is known that P g ≥ 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 : Z 1 −→ Z 2 is a finite map of degree n of Riemann surfaces, then any meromorphic function
on Z 1 satisfies a polynomial of degree n over the field of Z 2 . Hence
is a field extension of degree ≠ n.
f ∗ M(Z 2 ) ↩→ f ∗ (Z 1 )
Fact 2.11.1. The equality deg = n implies that all finite extensions of M(Z 2 ) are in natural bijective correspondence
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 Z 2 = P 1 , M(P 1 ) = C(Z) then there is a bijective correspondence between fields of
transcendence degree = 1 and finite covering of P 1 .
The equality deg = n implies that Aut(Z) = C-Aut of M(Z).
Question: We know that 2P g ≥ 2g. When are they equal?
If yes, then a loop C is homologous to 0 if and only if ∫ C ω = 0 for every ω ∈ Ω. Also, H1 dR
= Ω ⊕ Ω. The
values ∫ a i
ω and ∫ b i
ω are called periods.
31

2.12 Harmonic differentials and Hodge decompositions
Recall that A 1 = C ∞ -valued 1-forms.
Definition 2.12.1. Notice that locally every C ∞ -valued 1-form can be written as ω = fdz + gdz, where
fdz ∈ A 1,0 and gdz ∈ A 0,1 . There exists a C-linear map ∗ : A 1 −→ A 1 called the star operator, locally
defined by
∗(pdx + qdy) = −qdx + pdy
Or equivalently,
∗(fdz + gdz) = i(−fdz + gdz).
Every ω is uniquely a sum of ω 1,0 ∈ A 1,0 and ω 0,1 ∈ A 0,1 , and
∗(ω) = i(−ω 1,0 + ω 0,1 ).
Definition 2.12.2. ω ∈ A 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 if it is closed and coclosed.
Locally, ω = fdz + gdz is closed if g z = f z , and is coclosed if g z = −f z . Then if ω is harmonic we have
g z = f z = 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 H 1 be the space of harmonic differentials on Z. Then H 1 = Ω ⊕ Ω.
The Hodge Theorem states that H 1 = H 1 dR .
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). H 1 dR (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
( ) 1
ω − d
z n = ω + n dz
zn+1 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 principal
parts
∞∑
ω p = a i z i dz, when n ≥ 2.
i=−n
(2) There exist a meromorphic function with any prescribed value at a finite set of points.
(3) f ∗ M 2 ↩→ M 1 has degree = deg(f) for a finite map f : Z 1 −→ Z 2 .
33

2.13 Analysis on the Hilberts space of differentials
There exists a Hermitian inner product for 1-forms ω 1 and ω 2 (at least one which is compactly supported)
(ω 1 , ω 2 ) = ∫ Z ω 1 ∧ ∗ω 2 < ∞ .
Locally, ω i = p i dx + g i dy, with i = 1, 2. Then
ω 1 ∧ ∗ω 2 = (p 1 p 2 + g 1 g 2 )dx ∧ dy.
Hence we can define
∫
||ω|| 2 L = ω ∧ ∗ω < ∞.
2
Definition 2.13.1. Let B ′ be the space of bounded 1-forms ω such that ||ω|| 2 < ∞.
Z
With respect to the L 2 -norm, B 1 is a Hilbert space.
Let E be the closure in B 1 of dA 0 C , where A0 C is the space of C∞ -functions with compact support.
Theorem 2.13.1 (Orthogonal decomposition). Let ω ∈ B 1 . Then there exists a unique orthogonal decomposition
ω = ω h + df + ∗dg
where ω h is bounded harmonic and f, g ∈ A 0 , and df, dg ∈ E.
Proof: The essential point is that the space H is orthogonal to both E and ∗E, and E ⊥ ∗E.
ψ, ϕ ∈ A 0 then
∫
∫ ∫
〈 〉
dϕ, ∗dψ = − dϕ ∧ dψ = ψddϕ + d(ψdϕ)
Z
Z
Z
= 0 + 0.
If
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〉
= 〈ω, dϕ〉 .
Hence
H ⊕ ⊥ E ⊕ ⊥ ∗E ↩→ B 1 .
To show the equality, we go to the L 2 -completion of B ′ first.
Theorem 2.13.2 (Regularity). H = (Ě ⊕ ∗Ě)⊥ in ˇB 1 where ( ∨ ) means L 2 -completion.
34

Corollary 2.13.1 (Hodge decomposition). HdR 1 (Z) = Ω ⊕ Ω.
Proof: Saying that a form is closed means that it is orthogonal to ∗E. Hence the previous theorem
implies that a closed 1-form ω is uniquely ω h + df, i.e., every element of HdR 1 has a unique harmonic
representative.
35

2.14 Review
Theorem 2.14.1 (Orthogonal decomposition). For every ω ∈ A 1 = space of differntial 1-forms, there exists
a unique decomposition
ω = ω h + df + ∗dg
where ω h is a bounded harmonic, f, g ∈ A 0 = 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 = dA 0 and
∗E = ∗dA 0 . This fact follows easily using the L 2 -inner product and the equality ∗∗ = (−1) k , for
Riemann surfaces one has (−1) k = 1. We have
H⊥E⊥ ∗ E ⊆ A 1 .
Taking completion, we have
Ȟ⊥○Ě ⊕ ∗Ě = A1
where Ȟ⊥○Ě is the orthogonal (Ȟ ⊥ Ě) direct sum of Ȟ and Ě. By the Weyl’s Lemma, we have H = Ȟ.
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 = T f where
∫
T f [h] = hf
and h is compactly supported in U ⊂⊂ Z, where Z is a Riemann surface.
U
Corollary 2.14.1 (Hodge decomposition). HdR 1 (Z) = Ω ⊕ Ω = H.
36

2.15 Proof of Weyl’s Lemma
Claim 2.15.1. If f ∈ A 0 (C) then there exists ψ ∈ A 0 (C) such that ∆ψ = f.
Definition 2.15.1. A 0 c(U) = {f ∈ C ∞ (U) / f has compact support in U}.
Note that A 0 c(U) is a topological space of uniform convergence (not complete).
A distribution on U is a continuous linear functional T : A 0 c(U) −→ C.
Example 2.15.1. Let h ∈ A 0 c. Define
∫
T h [f] =
Then T is a distribution. Using integration by parts, for h ∈ A 0 c and f ∈ A 0 c, we have
∫
∫
hD α f = (−1) |α| fD α h,
where α = (α 1 , α 2 ), D α = ∂(α 1 +α 2 )
∂x α 1 ∂y α 2
U
hf
and |α| = α 1 + α 2 . In other words, we have
T D α h[f] = (−1) |α| T h [D α f]
Definition 2.15.2. (D α T )[f] = (−1) |α| T [D α f], for every f ∈ A 0 c.
Z
The definition for D α T h is the same as T D α h. Let g ∈ A 0 (U × I) where I is an interval, supp(g) ⊆ K × I
and K ⊂⊂ U.
Let f ɛ (z) = g(z,t+ɛ)−g(z,t)
ɛ
. Then f ɛ −→ ∂g
∂t
So:
Claim 2.15.2.
[ ]
d
dt T z[g(z, t)] = T ∂g(z,t)
z ∂t
.
as ɛ −→ 0. By continuity of T , we get
]
[
d
dt T z(g(z, t)) ←− ɛ→0
T [f ɛ ] −→ ɛ→0 ∂g(z, t)
T z
∂t
Similarly, if U and V are open in C and K ⊂⊂ U, L ⊂⊂ V , g ∈ A 0 (U × V ) are such that supp(g) ⊆ K × L,
then
[∫
] ∫
T z g(z, ɛ)d(Volɛ) = T z (g(z, ɛ))dɛ
V
V
Let ρ ∈ A 0 (C) satisfy supp(ρ) ⊆ D, ρ(z) = ρ 0 (|z|) for every z, and ∫ C ρ = 1. Set ρ ɛ(z) = 1 ɛ
ρ ( )
z
2 ɛ . Then
f ∈ A 0 (U) implies
∫
(ρ ɛ ∗ f)(z) = ρ ɛ (z − ɛ)f(ɛ)dVol(ɛ) ∈ A 0 (U (ɛ) )
where U (ɛ) = {z ∈ U / d(z, ∂U) > ɛ}.
U
37

Claim 2.15.3. For every α ∈ N 2 , we have
D α (ρ ɛ ∗ ρ) = ρ ɛ ∗ (D α f)
This result follows similarly applying a change of variables z ↦→ z + ɛ.
Claim 2.15.4. If z ∈ U (ɛ) and f is harmonic on D(z, ɛ), then
(ρ ɛ ∗ f)(z) = f(z).
Proof:
∫
(ρ ɛ ∗ f)(z) = ρ ɛ (ɛ)f(z + ɛ)dVol(ɛ) =
∫ 2π ∫ ɛ
ρ ɛ (r)f(z + rɛ iθ )rdrdθ = 2πf(z)
D(0,z)
0 0
0
∫ ɛ
ρ ɛ (r)rdr = f(z).
Proof of Weyl’s Lemma: Since ɛ −→ ρ ɛ (ɛ − z) has compact support in U, for every z ∈ U (ɛ) ,
h(z) := T ɛ [ρ ɛ (ɛ − z)]
is defined and belongs to A 0 (U (ɛ) ) by Claim 6.2. By Claim 6.4, it suffices to show that for every
f ∈ A 0 c(U (ɛ) ) we have
∫
T [f] = hf,
U (ɛ)
where h is harmonic since
∆h = T ɛ [∆ z ρ ɛ (ɛ − z)] = ∆T ɛ [ρ(ɛ − z)] = 0.
We want to show that T = T h . We have
Hence it suffices to show
∫
T [ρ ɛ ∗ f] = hf.
U (ɛ)
T [f] = T [ρ ɛ ∗ f].
By Claim 6.1, there exists ψ ∈ A 0 such that ∆ψ = f, where ψ is harmonic on V = C − supp(f). Hence
ψ = ρ ɛ ∗ ψ on V ɛ by Claim 6.4. Therefore, O = ψ − ρ ɛ ∗ ρ ∈ A 0 c(U) satisfies
since ∆T = 0, and T (∆ψ) = 0. Hence
∆O = ∆ψ − ρ ɛ ∗ ∆ψ = f − ρ ∗ f
T [f] = T [ρ ∗ f + ∆O] = T [ρ ɛ ∗ 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 there
exists a harmonic differntial ω on Z − {p} such that:
(1) ω − d ( 1
z n )
is harmonic on a small neighbourhood of p.
(2) ω ∈ B ′ Z−U , i.e., ω is bounded and smooth outside U, with ||ω||2 L 2 < ∞.
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
z n ∈ M(p).
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 h + df + ∗(dg)
and
ω = ψ − df = ω h + 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 ∈ A 1 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|| 2 Z−N = ||ω|| 2 + (dh, dh) + (ω, dh) + (ω, dh)
Z−N
= ||ω|| 2 Z−N + ||dh|| 2 Z−N
≥ ||ω|| 2 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.
Proof: f = (1, f 1 , f 2 , . . . , f n ) : Z −→ CP n if f(P ) = f(Q) for P ≠ Q. Then just adding a meromorphic
function 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 (P 1 , . . . , P L ), for
some polynomials P 1 , . . . , P l on P n .
( )
( )
Proof: Case n = 2: Consider f : Z −→ CP 2 and [z 0 , z 1 , z 2 ] ∈ C 3 . Then a = f ∗ z 0
z 2
and b = f ∗ z 1
z 2
are algebraic ( dependent ) over C, i.e., there exists a polynomial F (Z 1 , Z 2 ) of degree d such that F (a, b) = 0.
Then x d x 0
2F
x 2
, x1
x 2
defines f(Z).
This result is also known as the 1-st GAGA Principle (after Géométrie Algébrique et Géométrie 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 P n defined over a number field (i.e.,
(P 1 , . . . , P n ) = 0 defines a Riemann surface and P 1 , . . . , P n 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
map Z −→ CP 1 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.
Classification: π 1 (Z) = 0 if and only if the Riemann Mapping Theorem holds, Z ∼ = CP 1 , C or D.
Corollary 2.18.1. A Riemann surface Z is rational ( ∼ = CP 1 ) if and only if g(Z) = 0.
41

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.,
T x (M) := (T ∨ x (M)) ∨
Definition 3.1.1. An almost complex structure on an R-differentiable manifold X of dim R = 2n is an
epimorphism J of T X such that J 2 = −1. Or equivalently, it is the structure of a complex vector bundle on
T X.
A complex structure on X induces an almost complex structure on X by setting J = i = √ −1. We obtain a
map J : T X,R −→ T X,R with √ −1 acting on the domain and J acting in the codomain. We have
∂
∂z = 1 ( ∂
− i ∂ ) ( ) ∂ ∂
↦→ ,
2 ∂x i ∂y i ∂x i ∂y i
Locally, J is defined by
( ) ( ∂ ∂ ∂
, ↦→ , − ∂ )
∂x i ∂y i ∂y i ∂x i
corresponding to
the eigen-value i, an an eigen-value T 0,1
X
corresponding to the eigen-value −i, for the operator J. Note that
is naturally isomorphic to T X,R by taking the real part, and this isomorphism identifies i with J. Hence
is generated by vectors of the form u − iJu, with u ∈ T X,R.
Let (X, J) be an almost complex manifold. Then T X,R ⊗ C contains an eigen-bundle T 1,0
X
T 1,0
X
T 1,0
X
Theorem 3.1.1. A complex manifold has a complex structure J on T X,R and its associated subbundle
⊆ T X,R ⊗ C is naturally the same as T X (by taking the real part).
T 1,0
X
43

Similarly, if Ω X,R := T ∨ X,R , then
with the identification
Ω X,R ⊗ C = Ω X ⊕ Ω X
Ω X ←→ Ω 1,0
X ←→ dz i,
Ω X ←→ Ω 0,1
X ←→ dz 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 S 2 and S 6 . The 6-sphere S 6 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 S 2 ?
Theorem 3.1.2 (Newlande - Niremberg). J is integrable if and only if [T 1,0
X , T 1,0
X ] ⊆ T 1,0
X .
Proposition 3.1.1 (Poincaré Lemma). 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 (∂-Poincaré Lemma). 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 I,J dz I ∧ dz J
∂α = ∑ dα I,J ∧ dz I ∧ dz J = 0
Then α I = ∑ α I,J
∂z J is ∂-closed and of type (0, q), and so α I = ∂β if and only if α = (−1) p ∂( ∑ ∂z I ∧β I ).
Henceforth assume that α is of type (0, q), i.e., α = ∑ α J dz J . Apply the induction on the largest
integer k such that k ∈ J and α J ≠ 0. Necessarily k ≥ q and k = q implies α = fdz 1 ∧ · · · ∧ dz q . In the
latter case ∂α = 0 if and only if f is holomorphic in the variables z i with i > q. We may know apply
the following result:
Proposition: There exists a differentiable function g holomorphic in the variables z i , with i > q, such
that ∂g
∂z q = f and hence α = (−1)q−1 ∂(gdz 1 ∧ · · · ∧ dz q−1 ).
Now assume the ∂-Poincaré Lemma proved for k − 1 > q. Write α = α 1 + α 2 dz k , α 2 = ∑ α 2,J dz J where
|J| = q − 1 and J ⊆ {1, . . . , k − 1}. So ∂ = 0 implies α 2,J is holomorphic in variables z l , with l > k.
Hence by the previous proposition, we have α 2,J = ∂β 2,J
∂z k
, where β 2,J is holomorphic in z l , l > k. Then
∂β = ∂(β 2,J dz J ) = (−1) q−1 α 2 ∧ dz k + α ′ 1
44

where α ′ 1 involves only the coordinate z l for l < k. Thus,
α = α ′′
1 + ∂β
where α ′′
1 for l < k. Since β is holomorphic in the z l for l > k, we have ∂α ′′
1 = 0, q = deg(α ′′
1) < 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
g = 1 ∫
2πi C
f(z)
dζ ∧ dζ := lim
ζ − z ɛ→0
∫
1
2πi C\D(z,ɛ)
f(z)
dz ∧ dz.
ɛ − z
This limit exists since f is bounded and 1
ζ−z
in integrable on D. We want to show that ∂g = α = fdz.
∫
Now g = g ɛ where g ɛ = 1
f(γ+z)
2πi C\D(0,ɛ) γ
dγ ∧ dγ. Then
( ∫ )
∂g ɛ (z) = 1
dγ ∧ dγ
∂ z f(γ + z) dz
2πi C\D(0,ɛ)
γ
implies
∂
∂z g(z) = ∂ zg(z) = 1
2πi
∫
1
= lim
ɛ→0 2πi
∫
C
C\D(z,ɛ)
∂
∧ dγ
f(γ + z)dγ
∂z γ
∂
∂ζ
f(ζ)dζ
∧ dζ
ζ − z .
( )
On C\D(z, ɛ), we have ∂f dζ∧dζ
(z)
∂ζ ζ−z
= −d ɛ f(ζ) dζ
ζ−z
. By the Stokes Theorem, we get
Hence ∂g = fdz.
∫
1 ∂f dζ ∧ dζ
2πi C\D(z,ɛ) ∂ζ ζ − z = 1 ∫
f(ζ)
−→ f(z) as ɛ −→ 0.
2πi ∂D(z,ɛ) ζ − z
45

3.2 Kähler manifolds
Let V be a complex vector space with J = √ −1 and W = Hom R (V, R). Recall V C := V ⊗ C = V 1,0 ⊕ V 0,1 .
Hence also W C = W 1,0 ⊕ W 0,1 ⊇ W .
Definition 3.2.1. Let W 1,1 = W 1,0 ⊗ W 0,1 ⊆ ∆ 2 W C ⊇ ∆ 2 W ,
W 1,1 = {(1, 1)-forms} = {sesqui-linear forms on V }.
Let W 1,1
R
= W 1,1 ∩ ∆ 2 W = {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 1,1
H
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
1,1
H
∋ h ←→ Im(h) ∈ WR
Proof: Since h(u, v) = h(v, u), we have that Im(h) is alternating on V , i.e., Im(h) ∈ ∆ 2 W . 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 = ∑ i
2 a ijdz i ∧ dz j = −Im(h) ∈ Ω 1,1
X
∩ Ω2 X,R , where (a ij) is hermitian.
Definition 3.2.2. ω ∈ W 1,1
R
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 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 Kähler 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 Ω 2,0 and hence also on Ω 0,2 and its associated h is positive definite), then it is −Im(h) for a Kähler metric.
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
Kähler manifold.
Lemma 3.2.1. dVol h = ωn
n!
for (X, h), h = h ω = g(u, v), where ω n = ω ∧ · · · ∧ ω of type (n, n).
Proof: Let {e i } be an orthonormal basis of T X,x with respect to h. Then {e i , Je i } is a real orthonormal
basis for TX,x R ωn
with respect to g with positive orientation. It suffices to check
n!
= dx 1 ∧dy 1 ∧· · ·∧dx n ∧dy n
where
∑
{dx 1 , dy 1 , . . . , dx n , dy n } is the dual basis to (e i , Je i ). Let dz j = dx j + dy j . Then we have ω x =
j dz j ∧ dz j and
i
2
ω n x
n!
( ) n i ∏
= dz j ∧ dz j at x,
2
i
2 dz j ∧ dz j = dx i ∧ dy j .
j
Corollary 3.2.4. If X (n) is a compact Kähler manifold then [ω k ] ∈ HdR 2k (X) is nonzero for every k < n.
Proof: ω k = dγ implies ω n = d(ω n−k ∧ γ). The last implies 0 ≠ ∫ X ωn = 0, getting a contradiction.
Corollary 3.2.5. Let X (k) be a compact Kähler submanifold M. Then [x] ∈ H 2k (X) is nonzero.
Proof: Clearly h M | T X = h X and i ∗ ω (M,h) = ω (X,hX ), where i : X ↩→ M is the inclusion. If i(X) = ∂Γ,
then by the Stokes Theorem
∫ ∫
Volume X = i ∗ ωM h = dωM h = 0 since dωM h ≡ 0.
X
Γ
47

3.3 Metrics and connections
Let E −→ X be a C ∞ -vector bundle on X, and let A i (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
∇ : A 0 (E) −→ A 1 (E)
satisfying the Leibniz rule:
∇(fσ) = df ⊗ σ + f∇σ.
For a vector field ψ and σ ∈ A 0 (E) we write
∇ ψ σ = (∇σ)(ψ) ∈ A 0 (E).
In the case where E is a holomorphic vector bundle, we have the operation
∂ E : A 0 (E) −→ A 0,1 (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 unique
connection ∇ on T M called the Levi-Civita connection satisfying:
(1) d(g(ψ 1 , ψ 2 )) = g(ψ 1 , ∇ψ 2 ) + g(∇ψ 1 , ψ 2 ), i.e., g is ∇-invariant.
(2) ∇ ψ1 ψ 2 − ∇ ψ2 ψ 1 = [ψ 1 , ψ 2 ], 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 A 1 (E) −→ A 0,1 (E). Then ∇ 0,1 = ∂ E .
Theorem 3.3.2. The following statements for a complex Hermitian manifold (X, h) are equivalent:
(1) h is Kähler.
(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 T M 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
where Im(h) is a symplectic 2-form.
W 1,1
H
∼
−→ W 1,1
R
h ↦→ Im(h)
Theorem 3.4.2. The following conditions are equivalent for a complex Hermitian manifold (X, h):
(i) h is a Kähler metric, i.e., dw h = 0.
(ii) J is flat for the Levi-Civita connection of h.
(iii) The Chern connection of h on T 1,0
M equals the Levi-Civita connection on T R 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(ϕ 1 , ϕ 2 ) = ω(∇ϕ 1 , ϕ 2 ) + ω(ϕ 1 , ∇ϕ 2 ).
Let C ∞ (M) ∋ ϕ[ω(ϕ 1 , ϕ 2 )] = ω(∇ ϕ ϕ 1 , ϕ 2 ) + ω(ϕ 1 , ∇ ϕ ϕ 2 ). Since
dω(ϕ, ϕ 1 , ϕ 2 ) = ϕω(ϕ 1 , ϕ 2 ) − ϕ 1 ω(ϕ, ϕ 2 ) + ϕ 2 ω(ϕ 1 , ϕ) − ω([ϕ, ϕ 1 ], ϕ 2 )
the result follows from [ϕ i , ϕ j ] = ∇ ϕ1 ϕ j − ∇ ϕ2 ϕ 1 .
• (i) =⇒ (iii): The Chern connction equals the Levi-Civita connection for the flat metric ∑ i dz i ∧dz i .
The result follows from the following proposition.
Proposition 3.4.1. If (X, h) is a Kähler manifold and if x ∈ X, then there exists a holomorphic coordinate
(z 1 , . . . , z n ) centred at x such that
( ) ∂ ∂
h ij = h , = Im + O( ∑ |z i | 2 ).
∂z i ∂z j
The converse is also true.
49

3.5 The Fubini Study metric
Let L = {(l, v) ∈ CP n × C n / v ∈ l}. Consider the diagram
i π
L CP n 2
× C n+1 C n+1
π 1 ◦ i
π 1
CP n
The composite map π 2 ◦ i is called the blow up at 0. We have that L is a holomorphic line bundle over CP n
and is denoted by O(−1).
Definition 3.5.1. O CP n(h) := L −k where L −k := (L ∨ ) ⊗k for k > 0, is a holomorphic line bundle over CP 1 .
The standard metric ∑ |z i | 2 on CP n+1 restricts to a Hermitian metric on L. Its curvature (Ricci or Chern
form) is given by
ω = σ ∗ 2
2π ∂∂log|z i| i =
i
2π ∂∂log|σ|2
for any choice of a holomorphic section σ of L over CP n . Therefore, σ ′ = σf, for f ∈ O and so
where log|f| 2 = logf + logf and it is ∂∂-closed.
log(σ ′ ) 2 = log|σ| 2 + log|f| 2 ,
Lemma 3.5.1. ω is a positive (1, 1)-form.
Proof: We prove only the case n = 1. We have
So
ω =
∂log(1 + |z| 2 ) = ∂(1 + |z|2 )
1 + |z| 2 = zdz
1 + |z| 2 .
i [(1 + |z| 2 )dz ∧ dz − zdz ∧ zdz]
2π (1 + |z| 2 ) 2 = i dz ∧ dz
2π (1 + |z| 2 ) 2
and the conclusion follows from the transitivity of SU(n + 1) on T CP n .
Definition 3.5.2. ω is called the Fubini study metric in CP n and is denoted ω F S . It depends on the
choice of coordinates on C n+1 .
50

Similarly for a holomorphic vector bundle E −→ X with a Hermit metric h, one has the line bundle
i
L π −1 ˜π
(E) E
P(E)
π
X
Π
where P(E) = (E\{zero sections})/C ∗ and L is denoted by L = O P(E) (−1). The composition π ◦ i is called
the blow up of E at its zero section. Here π −1 (E) is the fibre product or pullback of π and Π. Let
F = (PE) x∈X be the fibre of π at x and f : F ↩→ PE the inclusion. Then f ∗ c 1 (| | 2 h
) is a positive (1, 1)-form,
where c 1 (| | 2 h ) = i
2π ∂∂log| |2 h . Hence c 1(| | 2 h
) is a (1, 1)-form on P(E) that is positive in the vertical direction
of π of X, where X is Kähler with Kähler form ω X . Hence P(E) is also Kähler.
Definition 3.5.3. O Eϕ (h) := L −k where L −k := (L ∨ ) ⊗k .
Note that given a vector bundle E −→ ϕ
X, then 1 ϕ = ϕ Eϕ (1) = L ∨ = L −1 is a holomorphic line bundle over
P(E).
Consider the compactification E = P(E ⊕ O) ⊇ E, E −→ 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)) and
hence it is Kähler.
ϕ
51

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 a
collection of objects F(U) of objects in A, for each open subset U ⊆ X, and a collection of morphisms
ρ UV : F(U) −→ F(V )
σ ↦→ σ| V = ρ UV (σ)
for each inclusion of open subsets V ↩→ U such that
ρ UV = ρ V W ◦ ρ UV .
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 s i ∈ F(U i )
be a collection of sections such that
s i | Uij = s j | Uij ,
where U ij = U i ∩ U j , then there exists a unique section s ∈ F(∪U i ) such that s| Ui = s i .
Definition 4.1.2. A morphism of (pre)sheaves is a map ϕ : F −→ G which associates to each open
subset U ⊆ X a morphism
such that for every V ⊆ U open
ϕ U : F(U) −→ G(U)
ρ UV ◦ ϕ U = ϕ V ◦ ρ UV . (compatibility)
Example 4.1.1. Sheaves of sections of vector bundles (C ∞ , C h , C ω for real analytic, O, etc).
53

Lemma 4.1.1. If F is a presheaf over X then there exists a unique sheaf F sh along with a morphism
F −→ F sh that factors thorough all morphisms F −→ G to a sheaf G.
F
G
ϕ ∃!
F sh
If X is a topological space, its structure sheaf CX 0 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 O X is given by the space of
regular functions on (Zariski) open subsets. A complex algebraic manifold is also a complex manifold, and
we write O alg
X
and Ohol X to distinguish the structures:
O alg
X
O hol
X
−→ Zariski topology,
−→ 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 open
set 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, tensor
products, 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.
isomorphic to A ⊕n is called locally free of rank n.
A sheaf that is locally
Remark 4.1.2. There exists a bijection between vector bundles of rank n and locally free sheaves of rank
n.
If A is a sheaf of fields, the rank one sheaves (invertible sheaves with respect to A) form a group under tensor
product, called the Picard group Pic OX (X).
Definition 4.1.5. Let (f, f # ) : (X, A) −→ (Y, B) be a morphism of ringed spaces, i.e., f : X −→ Y is a
continuous 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-module
G is defined as follows: set f (∗) G(U) = lim G(V ) and then set f ∗ G be the sheaf associated to the
−→f(U)↩→V
presheaf f (∗) G ⊗ f (∗) B A.
Example 4.1.2. The sheaf F of holomorphic sections of a holomorphic vector bundles F over X,
i X : {x} ↩→ X
(1) i (∗)
x F =: F x (the stalk of F at x and its elements are germs of sections of F at x).
54

(2) i ∗ xF = F x has finite dimension over C.
Definition 4.1.6. A sheaf of modules M on an algebraic variety (X, O X ) is said to be (quasi)-coherent
if 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 homomorphism
of 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 and
B 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 sheaf
of B-modules on Y given by V ↦→ F(f −1 (V )). Recall that for a sheaf of B-modules G on Y , its pullback is
defined as follows: Set f (∗) G(U) = lim G(V ), and then set (f ∗ G) to be the sheaf associated with the
−→f(U)⊆V
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 =: F x is called the stalk of i, and it equals the set of germs of sections of F .
(2) i ∗ F = F x , 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, O X ) is said to be quasicoherent
(resp. coherent) if it is locally isomorphic to the cokernel of a morphism of free shaves (resp. of
finite rank). By a free sheaf we mean a sheaf O ⊕n , where n is a cardinal number.
X
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 ∗ O alg
C
(2) i : C ∗ −→ C. Then i ∗ O alg
C ∗
= C[z] which is not finite dimensional.
= 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 {f 1 , . . . , f l } over C N .
We have
K[X] := C[z 1 , . . . , z N ]/ 〈f 1 , . . . , f l 〉
Then U ↦→ M ⊗ K[X] O X (U) is an O X -module and this correspondence preserves tensor product, exactness,
etc. Call this O X -module ˜M. In part, ˜M is quasi-coherent (and coherent if M is of finite type) and any
quasi-coherent O X -module is of this form.
Example 4.1.5 (Important). Ideal sheaf I Y
Y ↩→ X (i.e., Y is an algebraic subvariety).
↩→ O X corresponding to the subsheaf of O X vanishing on
We normally assume that the ground field K is algebraically closed. Then the Nullstellensatz tells us that
V (I) := sup(O X /I) ⊆ X is nonempty if and only if I ≠ O X . Hence the ringed space (V (I), O X /I) is
identified with the subscheme of X corresponding to I. A ringed space locally isomorphic to subschemes
of affine spaces is called al algebraic scheme.
Definition 4.1.9 (Associated fibre spaces). Let A be a quasi-coherent sheaf of a O X -algebra of finite type
(i.e., locally generated by finitely many sections as O X -algebras). We define a scheme S = Specm X A and
a morphism π : S −→ (X, O X ) as follows: Let X be affine. Then set
Specm X A := 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 = 0} c = a basis of open sets, form the Zariski topology.
Let π be the dual to the K-algebra homomorphism K[X] −→ A(X). If D(f) = {x ∈ X / f(x) ≠ 0}
for f ∈ O X (X) then by the quasi-coherence of A, we have A(D(f)) = A(X) ⊗ K[X] K[D(f)]. So that
SpecmA(D(f)) = π −1 (D(f)).
Special case: Given a coherent sheaf of O X -modules F, let A = Sym OX
F. Then the associated fibre space
is called the vector fibre space, denoted by π : V(F) −→ X. Here, Sym means the symmetric product
⊕ k≥0 (Sym k F). Note that
V(F) x = (F x /M x F x ) ∨
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[ɛ]/ɛ 2 ).
– The Zariski tangent space: Let x ∈ X be a point of al algebraic variety. Then T x X :=
(M x /M 2 x) ∨ .
– The normal bundle of Y ↩→ X (algebraic subvariety): Let I be the ideal sheaf of Y , then
I/I 2 = I ⊗ OX O Y . The normal vector bundle (or normal bundle) of Y in X is I/I . It is denoted
by N Y |X ( −→ π
Y ).
56

– The normal cone of Y in X is defined by
C Y |X := Specm Y
(
⊕k≥0 I k /I k+1) N Y |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 T f : T X −→ T Y . And it coincides with d x f : T x X −→ T f(x) Y , for every x ∈ X.
– We say that x ∈ X is a smooth point if C x (X) = T x X, 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 T X and such a form d gives a map
M x /M 2 x
where Ω ′ X = { differential on O X}.
∼
−→ Ω ′ X(x) := Ω ′ X,x/(Ω ′ X,x ⊗ M x )
(III) Blowing up a subscheme: Let I ↩→ O X be an ideal sheaf defining a subscheme Y ↩→ X, A =
⊕ k>0 I k . 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 algebra
A ⊗ OX O Y = ⊕ k≥0 I k /I k+1 , i.e., σ −1 (Y ) −→ Y is the projectivization of the normal cone C Y |N , i.e.,
0 = ⊗O X .
Definition 4.1.10. A sheaf is torsion free is ⊗O X = 0, i.e., it is supported on a subvariety.
Fact 4.1.2.
• Any torsion free O X -module F admits a resolution, i.e., a birational morphism σ : Y −→ X such that
σ ∗ F is locally free.
• (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)
˜X
σ
X
Y
57

4.2 Cohomology of sheaves
Let X be a topological space. We consider sheaves F i together with morphisms d i : F i −→ F i+1 such that
d i+1 ◦d i = 0 for every i. Such set of sheaves and morphisms (F i , d i ) 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 −→ F 0 such that i(F) = Ker(d 0 )
and Ker(d i+1 ) = Im(d i ), for every i.
(1) Čech resolution: Let F be a sheaf over X, {U i } i∈N a covering of X. For every finite subset I ⊆ N
set U I = ∩ i∈I U i , j I : U I ↩→ X and
Define F k := ⊕ |I|=k+i F I and d : F k −→ F k+1 by
F I = (j I ) ∗ (F| UI ) (extension by zero outside U I )
(dσ) j0...j k+1
= ∑ i
(−1) i σ j0 . . . ĵ i . . . j k+1 | U∩UI
where j 0 ≤ j 1 ≤ · · · ≤ j k+1 , σ = (σ I ), σ I ∈ F I (U) and |I| = k + 1. Lastly, we define ι : F −→ F 0 by
ι(σ) i = σ| U∩Ui
for σ ∈ F(U).
Proposition 4.2.1. This is a resolution.
(2) de Rham resolution: Let A k be the sheaf of C ∞ (R or C-valued) differential forms of degree k (on
a real or complex manifold). The d-Poincaré Lemma says that the complex (A k , d) is a resolution of
Ker(d 0 ) = R (resp. C), constant sheaves over X.
(3) Dolbeault resolution: Let E be a holomorphic vector bundle over a complex manifold X and E its
sheaf of holomorphic sections (i.e., E = O X (E)). Let A 0,q be the sheaf of C ∞ -sections of Ω 0,q
X ⊗ E.
Generalizing the d-Poincaré Lemma, we get that (A 0,q (E), ∂) is a resolution of Ker(∂ 0 ) = E = O X (E)
(a coherent O X -module).
58

4.3 Coherent sheaves
Let O X be the space of functions on X. The “sheaf version”of this space means that to every open subset
U ⊆ 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 poles
on 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 ⊕n
X
. An ideal sheaf I is just a subsheaf of O X which is an ideal
of O X (U) (a ring) for every U. Normally assume I ≠ O X . We have a short exact sequence
0 −→ I −→ O X −→ O Z(I) −→ 0,
where Z(I) is a scheme equal to Z(I) = spec(O X /I) X, which is nonempty. Note that O Z(I) and I are
examples of coherent sheaves over X, and O Z(I) is called the torsion part. Any coherent sheaf is locally a
finite direct sum of these factors. If there are no factors of the form O Z(I) , i.e., not supported on a proper
res
subvariety, then it is called torsion free. Any torsion free sheaf admits a resolution f : ˜X −→ X such
that f ∗ of the sheaf is locally free (i.e., a vector bundle). This is because given an ideal sheaf I defining a
subscheme Y ⊆ X, A = ⊕ k>0 I k is an algebra over O X . Then σ : X = proj(A) −→ X (an isomorphism
outside Y ), called the blowup of X along Y , is a birational map to X that replaces Y by a subscheme of
codimension 1, i.e., σ −1 (Y ) is locally given by one equation and σ −1 (Y ) −→ Y is the projectivization of the
normal cone C Y |X .
Given a collection of sheaves F i over X with morphism d i : F i −→ F i+1 such that d i+1 ◦ d i = 0 for every
i. It is a resolution of a sheaf F if there exists an inclusion i : F −→ F 0 such that j(F) = Ker(d 0 ) and
Ker(d i+1 ) = Im(d i ) for every i.
(1) There exists a sheaf X, {U i } i∈N covering of X. For every finite set I ⊆ N, set U I = ∩ i∈I U i , j I : U I ↩→ X,
and F I = (j I ) ∗ (F UI ) extended by zero outside U I . Set F k = ⊕ |I|=k+1 F I and d : F k −→ F k+1 by
(dσ) j0···j k+1
= ∑ i
(−1) i (σ j0···ĵ i···j k+1
)| U∩UI ,
where σ = (σ I ), σ I ∈ F k I (U), and j : F −→ F 0 is given by j(σ) i = σ| Ui∩U for σ ∈ F(U).
Proposition 4.3.1. This is a resolution, where F i | Ui
has trivial cohomology.
(2) de Rham resolution: Let A k be the sheaf of C ∞ (R or C)-valued k-forms. The Poincaré Lemma
says that the complex (A k , d) is a resolution of Ker(d 0 ) = R or C, constant sheaves.
(3) Let E be a holomorphic vector bundle and E its sheaf of holomorphic sections. Let A 0,q (E) be the
sheaf of C ∞ -sections Ω 0,q
X ⊗ E. The ∂-Poincaré Lemma implies that (A0,q (E), ∂) is a resolution of
Ker(∂ 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 R i F (M) for every i ≥ 0 in C ′ , unique up
to isomorphisms, satisfying the following conditions:
(1) R 0 F (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) −→ R 1 F (A) −→ · · ·
· · · −→ R i F (A) −→ R i F (B) −→ R i F (C) −→ R i+1 F (A) −→ · · ·
Remark 4.4.1.
(1) Let M 0 , i : A −→ M 0 be an acyclic resolution of A (i.e., R j+1 F (M k ) = 0 for every j, k ≥ 0). Then
R j F (A) = H j (F (M 0 )) := CoKer(d i−1 F(M i−1 ) −→ F(M i ))
(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 partition
of unity: for every open covering {U i }, there exists f i ∈ F(U i ) such that ∑ i f i = 1 (this sum is locally finite).
Proposition 4.4.1. H i (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) −→ A 0,q+1 (E))
Im(∂ : A 0,q−1 (E) −→ A 0,q (E))
Corollary 4.4.2 (Grothendieck Vanishing Theorem). Given a holomorphic vector bundle E −→ X, we
have H q (X, E) = 0 for q > n = dim C X.
Ker(d) = F −→ F 0 d
−→ F 1 −→ · · · −→ F k −→ · · ·
Consider a Čech resolution
where the map F −→ F 0 is given by σ ↦→ σ| U∩Ui and F k = ⊕ |I|=k+1 (j I ) ∗ (F| UI ).
60

Theorem 4.4.1. If H q>0 (U I , F) = 0 for every I ⊆ N then
H q (X, F) = Ȟq (U, F) := H q (Γ(F) = F(X), d X ),
e.g., if X is a finite dimensional manifold C ∞ , then there exists a good cover (U I contractible for every I)
and so
Ȟ q (U, Z) = H q cell
(X, Z)
where U is in an open cover and H q cell
(X, Z) denotes the cellular cohomology, which is computing via nerves
of the open covering.
61

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 ∆ ∗ TX,x ∨ . We assume X is oriented. Let α, β ∈ Ak : C ∞ (A k TX ∨ ). Then
∫
〈α, β〉 =
X
〈α, β〉 x
dVol(x)
gives an L 2 -metric on A k . We also have a pointwise isomorphism p : ∆ n−k Tx
∨ −→ Hom(∆ k Tx ∨ , ∆ n Tx ∨ ) given
by v ↦→ v ∧ −, where ∆ n Tx
∨ = RdVol(x), and an isomorphism m : ∆ k Tx
∨ ∼
−→ Hom(∆ k T ∨ , R) given by
e ↦→ 〈e, 〉 ∆ k T ∨.
∼
Definition 5.1.1. The Hodge Star Operator is given by
and the associated global isomorphism by
∗ = p −1 ◦ m : ∆ k T ∨ x
∗ : ∆ k T ∨
∼
−→ ∆ n−k T ∨ x
∼
−→ ∆ n−k T ∨
Ω k (X) −→ Ω n−k (X)
We extend ∗ to complex-valued forms by extending 〈 , 〉 to Hermit metrics on ∆ k C (T ∨ ⊗ C) = (∆ k R T ∨ ) ⊗ C.
We get (α, β) x dVol(x) = α x ∧ β x and so
∫
〈α, β〉 = α ∧ ∗β
is the L 2 -metric on A k C = Ak ⊗ C. In the case X is complex, A k C = ⊕ p+q=k ⊕ A p,q , A p,q = C ∞ (Ω p,q
X ).
X
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 L 2 metric in A k C .
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 vector
bundles given by h : Ω 0,q
X
⊗ E = Ω0,q (E) −→ (Ω 0,q ⊗ E) ∨ ∼ = Ω n,n−q ⊗ E ∨ , where ∆ 2n
X = Ωn,n = RdVol(x). So
it gives am antilinear isomorphism
∗ E : Ω 0,q (E)
∼
−→ Ω n,n−q (E ∨ ) = K X ⊗ Ω 0,n−q (E ∨ )
called the Hodge Star, where K X = Ω n,0 = ∆ n C T X ∨
bundle of a complex manifold X.
(holomorphic line bundle) is called the canonical
Fact 5.1.3. ∂ ∗ X = (−1) q ∗ −1
E ◦∂ K X ⊗E ∨ : A0,q (E) −→ A 0,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, ∆ d α〉 = ||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
{ (
⊕k Ω
(F, φ) = ( k X , ∆ d)
)
⊕q Ω 0,q (E), ∆ ∂E
for the Riemannian case,
for the Hermitian case.
where φ : F −→ F is an automorphism, and F = ⊕ k Ω k X or F = ⊕ qΩ 0,q (E).
65

5.2 Some applications of the Main Theorem
Theorem 5.2.1 (Riemannian case). Let H k be the space of harmonic k-forms. Then the map
H k −→ HdR k (X, R (or C)) given by α ↦→ [α], which is well defined since dα = 0, is an isomorphism.
Proof: By the Main Theorem, β ∈ A k 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 ∗ )) ⊥ . So
0 = (dγ, dd ∗ dγ) = ||d ∗ dγ|| 2 . Similarly, we get the analogous theorem for ∆ ∂E
where we can identify
H q (X, O(E) = E) with H 0,q (X, E) via the Dolbeaut isomorphism.
∂
Theorem 5.2.2. Given a holomorphic vector bundle E −→ X, let H 0,q (E) be the space of (∂ E )-holomorphic
forms of type (0, q) with values in E. Then the map H 0,q (E) −→ H q (X, E) given by α ↦→ [α] is an isomorphism.
Corollary 5.2.1. If X be a compact manifold then H q (X, R) is finite dimensional. If X is a compact
complex manifold and E −→ X is a holomorphic vector bundle then H q (X, E) is finite dimensional.
66

5.3 Review
Theorem 5.3.1. Let X be a compact manifold:
{ (⊕
k
(F, P ) =
Ωk X , ∆ )
( d
⊕ )
q Ω0,q (E), ∆ ∂E
if X is Riemannian,
E −→ X is a holomorphic vector bundle
where P is an elliptic or Laplacian-like operator. This implies that
where Ker(P ) is finite dimensional.
C ∞ (F ) = Ker(P )⊥○P (C ∞ (F ))
Corollary 5.3.1.
(1) There is an isomorphism H k ∼
−→ H k dR (X, Ror C) given by α ↦→ [α], where Hk is the finite dimensional
space of harmonic forms.
(2) There is an isomorphism H 0,q ∼
−→ H q (X, E), where H 0,q is the finite dimensional space of (0, q)-forms.
Both isomorphisms depend on the chosen metric.
Note that
HdR k (X, R) ∼= 1
Hk (X, R)
∼= 2 ∼ =
3
Ȟ(H, R)
where ∼ = 1 is given by the de Rham isomorphism, ∼ = 2 and ∼ = 3 are given by the Poincaré Lemma. Recall the
Dolbeaut isomorphism:
(X, E) ∼ = H 0,q (X, ∂ ∂
Ωp E) ∼ = (∗) H p,q (X, Ω p E) =: H p,q (X, E) (finite dimensional),
Ω p E ∼ = O(∆ p TX ∗ ⊗ E) (space of holomorphic p-forms with values in E).
H p,q
where (∗) comes from the isomorphism Ω p E = Ker∂ 0 (A p,0 )(E) −→ A p,1 (E)). Also,
H p,q (X) = H p,q (X, C) = H q (X, Ω p X
) (sheaf of holomorphic functions).
It is known that ∆ = ∂∆ ∂
in every Kähler manifold. Then the following theorem follows:
Theorem 5.3.2. Let X be a Kähler manifold. Then
H k (X, C) =
⊕
H p,q (X).
p+q=k
In other words, if [α] ∈ H k (X, C) with α harmonic, then α = ∑ p+q=k αp,q , where α p,q is the (harmonic)
component of type (p, q).
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 S 1 , and in general for the torus as follows:
F (0, t) = ∑ a n (t)e inθ .
We have
∂ t + ∆ θ = 0 =⇒ 0 = ∑ (
a
′
n (t) + n 2 a n (t) ) e inθ
=⇒ a n (t) = a n e −n2t , where a n = a n (0),
=⇒ F (θ, t) = ∑ e −n2t a n e inθ .
n≥0
It follows F (θ, t) −→ a 0 = ∫ S 1 f(θ)dθ = Av S 1(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 ∫
∫
e − (x−y)2
4t f(y)dy = e R (x, y, t)f(y)dy.
4πt
R
The function e R (x, y, t) = 1 √
4πt
e − (x−y)2
4t is called the heat kernel. Similarly, e R n(x, y, t) = 1 √
4πt
e − ||x−y||2
4t .
On S 1 , we have e(x, y, t) = ∑ n e−n2t e in(x−y) = ∑ n e−n2t e inx e iny , where e −n2t are the eigenvalues of ∆, and
e inx and e iny are the eigenfunctions. In general, the existence of e X (x, y, t) is difficult to obtain analytically
but trivial on physical grounds.
R
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) = 0
(∗)
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 (∗).
Proof: ∂ t ||A(t)|| 2 = 2 〈∂ t A, 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 K p (x, y, t) ∈ A p x(X),
depending only on (X, g) and p, called the heat kernel of X on p-forms such that
∫
A(t) = K p (·, y, t)α(y)dVol(y)
solves (∗), for every α ∈ A p (X).
X
Let T t (x) = ∫ K(·, y, t)α(y)dy.
X
Theorem 5.4.2. T t satisfies:
(1) T t1+t 2
= T t1 T t2 .
(2) T t is formally self-adjoint.
(3) T t α tends to a C ∞ harmonic form H(α) as t −→ ∞.
(4) G(α) = ∫ ∞
0 (T tα − 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(t 1 + t) solves the heat equation with initial condition A(t 1 ).
(2) ∂ t 〈T t η, T τ ɛ〉 = 〈∂ t T t η, T τ ɛ〉 = − 〈∆T t η, T τ ɛ〉 = − 〈T t η, ∆T τ ɛ〉 = 〈T t η, ∂ t T τ ɛ〉 = ∂ t 〈T t η, T τ ɛ〉. This
implies that 〈 , 〉 is a function of t + τ, so denote 〈 , 〉 by g(t + τ). Therefore,
(3) (1) + (2) =⇒ ∀h > 0,
〈T t η, ɛ〉 = g(t + 0) = g(0 + t) = 〈η, T τ ɛ〉 .
||T t+2h α − T t α|| 2 = ||T t+2h α|| 2 + ||T t α|| 2 − 2 〈T t+2h α, T t α〉
= ||T t+2h α|| 2 − ||T t α|| 2 − 2(||T t+h α|| 2 − ||T t+2h α||||T t α||)
and ||T α α|| 2 converges, and therefore it is decreasing. Hence ||T t+2h α − T t α|| 2 −→ 0. It follows
T t α −→ H(α), for some H(α) ∈ A p (X) L2 , called the harmonic projection. Fix τ > 0, then
T t α = T τ T t−τ α −→ H(α) := T τ H(α) as t −→ ∞. Hence H(α) is C ∞ since T τ is given by a C ∞
kernel. Hence H = lim t→∞ T t is also formally self-adjoint.
(4) ||T t α − Hα|| can be shown to decay rapidly enough so that G(α) = ∫ ∞
0 (T tα − H(α))dt is well
defined. We verify that G is formally the Green operator:
∆G(α) =
and for β harmonic we have
〈Gα, β〉 =
∫ ∞
0
∫ ∞
〈(T t − H)α, β〉 dt =
0
∆T t αdt =
∫ ∞
0
∫ ∞
0
−∂ t T t αdt = α − H(α),
〈α, (T t − H)β〉 dt = 0, since (T t − H)β = 0.
69

Corollary 5.4.1. There exists an orthogonal direct sum decomposition
where Im(∆) = Im(d) + Im(d ∗ ).
A p (X) = H p (X) ⊕ d(A p−1 (X)) = H p (X) ⊕ d ∗ (A p−1 (X))
Corollary 5.4.2. There is an isomorphism H p (X) ∼ = H p dR
(X) given by α ↦→ [α].
Then O ∈ A p (X) =⇒ O = α + dd ∗ γ + d ∗ dγ, where α ∈ H(X). Also,
||d ∗ dγ|| 2 = 〈d ∗ dγ, α〉 = 〈dγ, dα〉 , where dα = 0.
70

5.5 Index Theorem (Heat Equation approach)
Let ∆ p : A p (X) −→ A p (X), λ ∈ R ≥0 . Let E p λ denote the λ-eigenspace for ∆p (finite dimensonal). The
square root √ ∆ = δ is called the Dirac operator.
Lemma 5.5.1. The sequence
is exact for λ > 0.
0 −→ E 0 λ
d
−→ Eλ 1 d
−→ · · · −→ Eλ n −→ 0
Proof: ω ∈ E p λ =⇒ ∆p+1 dω = d∆ p ω = λdω =⇒ dω ∈ E p+1
λ
.
Now ω ∈ E p λ and dω = 0 =⇒ λω = ∆p ω = d ∗ d + dd ∗ ω =⇒ ω = d ( 1
λ d∗ ω ) . Then ∆d ∗ ω = d ∗ ∆ω = λd ∗ ω.
Corollary 5.5.1. ∑ p (−1)p dim(E p λ ) = 0.
Corollary 5.5.2. Let {λ p i } be the spectrum of ∆p , with terms repeated n times if multi = n. Then
where ∑ ′
i
is over i where λ(p) i = 0.
∑
(−1) p tre −t∆p = ∑
p
p
(−1) p e −tλ(p) i
= ∑ p
(−1) p ′∑
i
e −λ(p) i (t)
Note that ∑ ′
i e−λ(p) i
(t) = dim(Ker(∆ p )). Hence
X (X) = ∑ p
= ∑ p
(−1) p dim(Ker(∆ p )) = ∑ (−1) p tre −t∆(p) , where e −t∆(p) = T t ,
p
(−1) ∑ ∫
p e (p) (x, x, t)dVol(x).
i X
Proposition 5.5.1. e(x, x, t) ∼ (4πt) −n/2 ∑ ∞
k=0 u k(x, t)t k , where u k (x, t) is explicitly given in terms of
components of curvatures.
Hence, as t −→ 0, we have
(
X ∼ 1 n/2 ∑
∞ ∫
4πt
k=0
X
)
∞∑
(−1) p tru p k (x, x)dVol(x) t k .
p=0
71

This implies
4π n/2 ∫
∞∑
{
(−1)tru p 0
k (x, x)dVol(x) = X (X)
X p=0
if k ≠ n/2,
k = n/2.
Theorem 5.5.1 (Gauss-Bonet). Let n = dim(X) be even. Then
∫
X (X) = ω,
where ω is given in a local frame by
∑
ω = c n (signσ)(signτ)R σ(1)σ(2)τ(1)τ(2) · · · R σ(n−1)σ(n)τ(n−1)τ(n)
and c n =
(−1)n/2
(8π) n/2 ( . n 2 )!
σ,τ
X
For n = 2, ω = 1
8π (R 1212 −R 1221 −R 2112 −R 2121 )dA = − 1
2π RdA 1212 = K 2π
dA where K is the Gaussian curvature.
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.
73