Math 259, Winter 2014 - Riemann surfaces and algebraic
Dragos Oprea, MW 10-11:20, APM 7-281
Topics to be covered:
1. Complex manifolds. Riemann surfaces. Basic definitions. Examples.
2. Sheaves and their cohomology. Cech cohomology. Dolbeault cohomology.
3. Divisors and line bundles. Linear systems and projective embeddings.
4. The Riemann-Roch theorem and applications.
5. Serre duality.
6. Curves of low genus.
7. An introduction to the moduli space of curves.
Prerequisities: Basic complex analysis. Some differential
geometry and some algebraic geometry are useful, but not formally
required. The best approach to see if you have the prerequisites is to
attend the first few lectures.
- Lecture 1:
Sheaves. Ringed spaces. Complex manifolds. Coordinate charts.
- Lecture 2: Examples: projective line, elliptic
projective curves. Holomorphic and meromorphic functions
on Riemann surfaces. Meromorphic functions on the projective line.
Meromorphic functions on
tori via theta functions.
- Lecture 3: Holomorphic maps are open. Global structure of
holomorphic maps and branched covers. Holomorphic functions have the
same number of zeros and poles.
- Lecture 4: The tangent sheaf. The tangent sheaf of a
complex manifold is locally free. Vector
bundles and examples. Algebraic
constructions with vector bundles. Equivalence between vector bundles and
locally free sheaves. Line bundles and divisors. Rational
- Lecture 5: Almost complex
manifolds. Decomposition of differential
forms into bitypes. Newlander-Nirenberg theorem. Conformal
structures and Riemann surfaces.
- Lecture 6: Functorial properties of sheaves. Kernel, cokernel, image, short exact
sequence. Sheafification. Examples. Flabby sheaves. Sheaf of
- Lecture 7: Properties of flabby sheaves. Canonical flabby
resolution. Cohomology of sheaves. Acyclic resolutions. Abstract deRham
- Lecture 8: Soft sheaves. Flabby sheaves are soft. Soft
sheaves over paracompact spaces. Fine sheaves. Sheaves of C^infinity
modules are fine. Application: deRham theorem.
- Lecture 9: The Dolbeault resolution and
Dolbeault cohomology. The inhomogeneous Cauchy-Riemann equation
for compact supports.
- Lecture 10: The inhomogeneous Cauchy-Riemann equation with
Cohomology of open sets in C with coefficients in the sheaf of holomorphic
functions, and the sheaf of nowhere zero holomorphic functions. The
Weierstrass and Mittag Leffler theorems.
- Lecture 11: Cech cohomology. Examples. Leray covers. The
equivalence between Cech cohomology and the flabby cohomology.
- Lecture 12: General discussion of Cartan-Serre finitness
theorem, Serre duality, Kodaira vanishing, Riemann-Roch. Adapted coverings
for vector bundles compute Cech cohomology. Bounded Cech cohomology.
- Lecture 13: Cartan-Serre finiteness theorem and its proof.
Statement of Serre duality. Review of topologies on various spaces of
- Lecture 14: The sheaf of distributions. Dolbeault
resolution with distribution coefficients. Proof of Serre duality.
- Lecture 15: General discussion of line bundles.
Cohomological classification of line bundles. Chern forms and Chern
- Lecture 16: Comparison between algebraic and differential
geometric approaches to Chern classes. The degree of a line bundle is
obtained by integrating the Chern form.
- Lecture 17: The number of zeros and poles of a
holomorphic section equals the
degree of the line bundle. Meromorphic sections. Every line bundle comes
from a divisor. Riemann-Roch theorem. The
degree of the canonical bundle. The genus is a topological
invariant.The cohomology of line bundles over a genus
g surface in terms of the degree. Clifford's theorem.
- Lecture 18: Projective embeddings. Globally generated,
ample and very ample line bundles. Criterion for basepointfreeness
and very ampleness. Quantitative study of these notions on Riemann
surfaces based on
degree. Every compact Riemann surface is projective. The canonical
- Lecture 19: Hyperelliptic curves. Canonical morphism for
hyperelliptic and non-hyperelliptic curves. Classification of curves of
low genus: genus 0, genus 1 as plane cubics and also as intersection
in 3-space, genus 2 curves are hyperlliptic, genus 3 curves are
- Lecture 20: Classification of curves as genus 4 and genus
5. The moduli space of stable curves and sketch of the construction of the
moduli space. Kodaira
dimension of the moduli space of curves and the Eisenbud-Harris-Mumford
Homework 1 due January 22, PDF
Homework 2 due February 3, PDF