Math 244, Spring 2008 - Riemann surfaces and algebraic
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 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: 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
sheaf of a complex manifold
is locally free.
- Lecture 5: The sheaf of differential forms and its
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.
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:
sheaves. The cannonical flabby resolution. Sheaf cohomology defined.
- Lecture 9: Acyclic resolutions. Abstract de Rham
- Lecture 10: Soft sheaves are
Fine sheaves. Sheaves of C^infty modules are fine. Applications:
theorem. The Dolbeault resolution and the inhomogeneous
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
- Lecture 13: Cech cohomology.
The equivalence between Cech cohomology and the flabby cohomology.
- Lecture 14: General discussion of Cartan-Serre
theorem, Serre duality, Kodaira vanishing, Riemann-Roch. Adapted coverings for vector bundles compute Cech
cohomology. Bounded Cech cohomology.
15: Finite dimensionality of cohomology.
16: Differential forms with distribution coefficients. Serre
- Lecture 17: Proof of Serre duality.
resolutions for forms with distribution coefficients. Baby version of
- Lecture 18: Line bundles and
classes. Chern forms.
- Lecture 19: I showed that
associated to divisors, the Chern form integrates to the degree of the
- Lecture 20: Any holomorphic section has
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
- Lecture 21: Riemann-Roch and its proof.
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
Jacobian and Abel-Jacobi maps
for arbitrary curves.
- Lecture 24: Linear series. Basepointfree, ample and
line bundles. Projective embeddings. Quantitative study of these notions
in terms of degree.
- Lecture 25: Genus 1 curves can be viewed as cubics in
intersection of two quadrics in P^3. Hyperlliptic curves.
- Lecture 26: Genus 2 curves are hyperelliptic. Canonical
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
- Lecture 27: Moduli space of genus g curves. Fine vs.
representability. The Deligne-Mumford compactification.
- Lecture 28: A discussion of the Picard group of the
curves. The canonical bundle. A
discussion of the Kodaira dimension of the moduli space.