Math 259, Winter 2014 - Riemann surfaces and algebraic curves

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 Summaries

• Lecture 1: Presheaves. Sheaves. Ringed spaces. Complex manifolds. Coordinate charts. Riemann surfaces.
• Lecture 2: Examples: projective line, elliptic curves, 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 equivalence.
• 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 discontinuous sections.
• Lecture 7: Properties of flabby sheaves. Canonical flabby resolution. Cohomology of sheaves. Acyclic resolutions. Abstract deRham theorem.
• 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 arbitrary supports. 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 functions.
• 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 classes.
• 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 morphism.
• 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 of quadrics in 3-space, genus 2 curves are hyperlliptic, genus 3 curves are plane quartics.
• 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 theorem.

Homework:

• Homework 1 due January 22, PDF
• Homework 2 due February 3, PDF