Math 244, Spring 2008 - Riemann surfaces and algebraic curves

Dragos Oprea, MWF 10-10:50, 381 T

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. Riemann-Hurwitz formula.

7. Curves of low genus.

8. The Jacobian. The Abel-Jacobi map.

9. 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: Vector bundles and examples. Algebraic constructions with vector bundles. Equivalence between vector bundles and locally free sheaves.
• Lecture 4: Vector fields. The tangent sheaf. The tangent sheaf of a complex manifold is locally free.
• Lecture 5: The sheaf of differential forms and its splitting via bi-types. A discussion of almost complex manifolds and Newlander-Nirenberg theorem, integrable and involutive almost complex structures.
• Lecture 6: Line bundles associated to divisors. Linear equivalence of divisors. Classification of line bundles over the projective line and tori.
• Lecture 7: Sheaf theory. Kernels and cokernels of sheaf morphisms. Sheafification. Exact sequences of sheaves. Examples: exponential sequence, ideal sheaf sequence, the sheaf of divisors. Resolutions of the constant sheaf by differential forms and Poincare lemma.
• Lecture 8: Flabby sheaves. The cannonical flabby resolution. Sheaf cohomology defined.
• Lecture 9: Acyclic resolutions. Abstract de Rham theorem. Soft sheaves defined.
• Lecture 10: Soft sheaves are acyclic. Fine sheaves. Sheaves of C^infty modules are fine. Applications: de Rham theorem. The Dolbeault resolution and the inhomogeneous Cauchy-Riemann equation.
• Lecture 11: The inhomogeneous Cauchy-Riemann equation for compact supports. The inhomogeneous CR for open subsets of C without restrictions on supports. Dolbeault cohomology of polydisks.
• Lecture 12: Applications of the inhomogeneous CR. 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 13: Cech cohomology. The equivalence between Cech cohomology and the flabby cohomology.
• Lecture 14: 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 15: Finite dimensionality of cohomology.
• Lecture 16: Differential forms with distribution coefficients. Serre duality stated.
• Lecture 17: Proof of Serre duality. Dolbeault resolutions for forms with distribution coefficients. Baby version of elliptic regularity.
• Lecture 18: Line bundles and Chern classes. Chern forms.
• Lecture 19: I showed that for bundles associated to divisors, the Chern form integrates to the degree of the divisor.
• Lecture 20: Any holomorphic section has as many zeros as the degree of the line bundle. Any line bundle is the line bundle of a divisor. Any meromorphic function has as many zeros as poles.
• Lecture 21: Riemann-Roch and its proof. The cohomology of line bundles over a genus g surface in terms of the degree. Clifford's theorem.
• Lecture 22: The degree of the canonical bundle. The genus is a topological invariant. Riemann-Hurwitz. Curves of low genus. Genus 0.
• Lecture 23: Genus 1 curves and Jacobians. More details on the Jacobian and Abel-Jacobi maps for arbitrary curves.
• Lecture 24: Linear series. Basepointfree, ample and very ample line bundles. Projective embeddings. Quantitative study of these notions in terms of degree. Canonical morphisms.
• Lecture 25: Genus 1 curves can be viewed as cubics in P^2 or intersection of two quadrics in P^3. Hyperlliptic curves.
• Lecture 26: Genus 2 curves are hyperelliptic. Canonical maps of hyperelliptic and non hyperelliptic curves. Genus 3 nonhyperelliptic are quartics in P^2. Genus 4 nonhyperelliptic is an intersection of quadric and cubic in P^3. Genus 5 are either hyperelliptics, trigonal curves which are on a Hirzebruch surface, intersetions of 3 quadrics.
• Lecture 27: Moduli space of genus g curves. Fine vs. coarse representability. The Deligne-Mumford compactification.

• Lecture 28: A discussion of the Picard group of the moduli of curves. The canonical bundle. A discussion of the Kodaira dimension of the moduli space.