Riemann surfaces Pablo Arés Gastesi

219 Pages·2006·1.3 MB·English

Riemann surfaces
Pablo Ar´es Gastesi
School of Mathematics, Tata Institute of Fundamental Research,
Bombay 400 005, India
[email protected]
ii

Acknowledgements
I would like first of all to thank V. Srinivas for his continuous encouragement to
write this book.
I would also like to thank the people who attended my course, on which these notes
are based: Pralay Chatterjee, Preeti Raman and Vijaylaxmi Trivedi.
Finally I would like to thank R.R. Simha for many comments, all of them very
helpful.
Contents
Acknowledgements iv
Chapter 1. Riemann surfaces 1
1.1. Background 2
1.2. Topology of Compact Orientable Surfaces 12
1.3. Riemann Surfaces and Holomorphic Mappings 17
1.4. Differential Forms 28
1.5. Sheaf Cohomology 42
Chapter 2. Compact Riemann surfaces 57
2.1. Divisors 58
2.2. Dolbeault’s Lemma and Finiteness Results 62
2.3. The Riemann-Roch Theorem 68
2.4. Line bundles and Divisors 71
2.5. Serre Duality 81
2.6. Applications of the Riemann-Roch Theorem 95
2.7. Projective embeddings 99
2.8. Weierstrass Points and Hyperelliptic Surfaces 106
2.9. Jacobian Varieties of Riemann Surfaces 115
Chapter 3. Uniformization of Riemann surfaces 127
3.1. The Dirichlet Problem on Riemann surfaces 128
3.2. Uniformization of simply connected Riemann surfaces 141
3.3. Uniformization of Riemann surfaces and Kleinian groups 148
3.4. Hyperbolic Geometry, Fuchsian Groups and Hurwitz’s Theorem 162
3.5. Moduli spaces 178
Exercises 187
v
vi CONTENTS
Bibliography 201
Notation 203
Index 206
List of Figures 213
CHAPTER 1
Riemann surfaces
1.1 Background 2
1.2 Topology of Compact Surfaces 12
1.3 Riemann Surfaces and Holomorphic Mappings 17
1.4 Differential Forms 28
1.5 Sheaf Cohomology 42
1
2 1. RIEMANN SURFACES
1.1. Background
This section is divided into three parts. In the first one ( 1.1.1 to 1.1.13) we
§§
review some results of Complex Analysis that we need in this book. Some of those
results will be extended later to the setting of Riemann surfaces. We also introduce
a few results from the theory of normal families of holomorphic functions, a topic
that might not be very familiar to some readers.
The second part( 1.1.14 to 1.1.23)contains basic concepts andresults of Topol-
§§
ogy: fundamental groups, covering spaces and partitions of unity.
Inthelast part( 1.1.24and1.1.25)westateatheoremofSchwartz onoperators
§§
on Banach spaces that we need in section 2.1.
References for the results in this section are given in 1.1.26.
§
Complex Analysis
1.1.1. Definition. Let Ω be an open subset of the complex plane. A complex
valued function f : Ω C is holomorphic at a point z Ω if the complex
0
→ ∈
derivative
f(z) f(z )
0
f (z ) = lim −
′ 0
z→z0 z −z0
exists. We say that f is holomorphic in Ω if it is holomorphic at every point of Ω.
1.1.2. For a complex valued function f with partial derivatives, ∂f/∂x and
∂f/∂y,wedefinethecomplexderivativesf andf bytheexpressions(seealso1.4.1),
z z¯
∂f 1 ∂f ∂f ∂f 1 ∂f ∂f
f = = i , f = = +i .
z z¯
∂z 2 ∂x − ∂y ∂z¯ 2 ∂x ∂y
(cid:18) (cid:19) (cid:18) (cid:19)
Informallyspeaking,thenextresultsaysthatafunctionisholomorphicif“itdepends
on z but not on z¯”.
Proposition. Let f : Ω C be a function with continuous partial derivatives
→
at all points of Ω. Then f is holomorphic if and only if f = 0 (in Ω). If f is given
z¯
by f(z) = u(z)+iv(z), where u and v are real valued functions defined on Ω, then
1.1. BACKGROUND 3
f is holomorphic if and only if it satisfies the Cauchy-Riemann equations:
∂u ∂v ∂u ∂v
= , = .
∂x ∂y ∂y −∂x
The functions u andv above arecalled the realand imaginaryparts off. Sometimes
we will use the notation Re(f) and Im(f) for u and v respectively.
1.1.3. We record for later use (2.2.4) how the operators ∂ and ∂ behave
∂z ∂z¯
under composition [18, pg. 31]. Assume f : Ω C and g : Ω C are holomorphic
′
→ →
functions with f(Ω) Ω. Then
′
⊂
(g f) = (g f)(f )+(g f)(f )
◦ z z ◦ z z¯◦ z¯
(g f) = (w f)(f )+(g f)(f ).
◦ z¯ z ◦ z¯ z¯◦ z
1.1.4. Proposition. A function f : Ω C is holomorphic if and only if
→
for every point z Ω, there exists a neighbourhood U Ω of z , such that f can
0 0
∈ ⊂
be written as a convergent power series f(z) = ∞n=0an(z −z0)n in U. If f is
holomorphic, then f′(z) = ∞n=1ann(z −z0)n−1. P
P
Corollary. Holomorphic functions are smooth (of class C ) that is, they have
∞
derivatives of all orders.
We actually have that holomorphic functions are of class Cω; that is, they are
representable by power series (there exist functions with derivatives of all order
which are not given by power series).
The following result simplifies the local expression of holomorphic functions.
1.1.5. Proposition. If f : Ω C is a non-constant holomorphic function,
→
then for every point z Ω, there exists a positive integer n, and a neighbourhood
0
∈
U of z in Ω, such that f(z) = (z z )ng(z), where g : U C is a holomorphic
0 0
− →
function satisfying g(z ) = 0.
0
6
The integer n is called the order of the zero of f(z) f(z ) at z .
0 0
−
4 1. RIEMANN SURFACES
1.1.6. Theorem (Identity Principle). Let f,g : Ω C be two holomorphic
→
functions defined on a connected, open subset Ω of C. Assume that there exists a
sequence z of points of Ω, with z z , where z Ω, such that f(z ) = g(z ),
{ n}n n →n 0 0 ∈ n n
for all n. Then f(z) = g(z) for all z Ω.
∈
Remark. There are two important observations to be made about the Identity
Principle: first of all, the set Ω is assumed to be connected; otherwise, the result
will hold only in the connected component of Ω containing the sequence z and
n n
{ }
the limit point z . Secondly, the limit point z must belong to the set Ω; there exist
0 0
examples of distinct holomorphic functions that take the same values at each point
of a convergent sequence of points of an open set, where the limit point of such
sequence does not belong to the open set.
Recall that a subset A of a topological space X is called discrete if A does not have
accumulation points in X (A = ).
′
∅
Corollary. Let f : Ω C be a non-constant holomorphic function. Then for
→
any complex number a, the set z Ω; f(z) = a is discrete in Ω (it does not have
{ ∈ }
limit points in Ω).
1.1.7. Denote by D the unit disc, D = z C; z < 1 .
{ ∈ | | }
Lemma (Schwarz). Let f : D C be a holomorphic function satisfying f(0) = 0
→
and f(z) 1. Then
| | ≤
(1) f(z) z ;
| | ≤ | |
(2) f (0) 1.
′
| | ≤
Moreover, if equality holds in (1) for a point z = 0, or in (2), then f(z) = λz, for a
6
complex number λ satisfying λ = 1.
| |
1.1.8. A point z C is called an isolated singularity for a function f if f
0
∈
is defined and holomorphic on U z , where U is a neighbourhood of z . Isolated
0 0
\{ }
singularities are of three types:
(1) removable if f can be extended to a holomorphic function in U;
(2) pole if there exists a positive integer n, such that (z z )nf(z) can be
0
−
extended to a holomorphic function in U;
(3) essential singularity if none of the above conditions are satisfied.