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

Course Information PDF

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. Canonical maps. Classification of curves of low genus.

8. The Jacobian. The Abel-Jacobi map.

9. An introduction to the moduli of curves.

Lecture Summaries

• Lecture 1: Presheaves. Sheaves. Ringed spaces. Real and complex manifolds. Coordinate charts.
• Lecture 2: Examples of Riemann surfaces. The projective line. Tori. Projective curves. Holomorphic and meromorphic functions on Riemann surfaces. Meromorphic functions on the projective line. Meromorphic functions on tori via theta functions. Showed that the Jacobi theta function has a unique simple zero.
• Lecture 3: Meromorphic functions on the torus are ratios of products of theta functions. Incidentally, I defined divisors on Riemann surfaces and showed that on a torus or on P^1, the divisor associated to a meromorphic function has degree 0. I started speaking about vector fields. I defined the tangent sheaf.
• Lecture 4: I defined the tangent space. I showed that the tangent sheaf is locally free. Equivalence between locally free sheaves and vector bundles. For a complex manifold, I defined the (1,0) and (0,1) vector fields and differential forms.
• Lecture 5: I showed that the sheaf of differential forms has a further splitting depending on 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. On P^1, the line bundle L(D) is determined by deg (D). On a torus, the line bundle L(D) is determined by deg D and the sum a(D).
• Lecture 7: Kernels, cokernels and images 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 (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. Examples: the deRham theorem, Dolbeault resolution. Started the proof of the Dolbeault resolution by solving the inhomogeneous Cauchy-Riemann equation. Started with the case of compactly supported functions.
• Lecture 11: The inhomogeneous Cauchy-Riemann equation for compact supports. Dolbeault resolutions for arbitrary dimension. The inhomogeneous Cauchy-Riemann equation for open subsets of C without restrictions on supports. Dolbeault cohomology of polydisks.
• Lecture 12: As an application of the inhomogeneous CR, I showed that cohomology of any open set in C with coefficients in the sheaf of holomorphic functions vanishes. I showed that that implies the same about the sheaf of nowhere zero holomorphic functions. I derived the Weierstrass and Mittag Leffler theorems. Started Cech cohomology.
• Lecture 13: Showed equivalence between Cech cohomology and the flabby cohomology.
• Lecture 14: General discussion of Cartan-Serre finitness theorem, Serre duality, Kodaira vanishing, Riemann-Roch. I showed that H^0 is finite dimensional. Adapted coverings for vector bundles compute Cech cohomology in dimension 1. Bounded Cech cohomology.
• Lecture 15:Finite dimensionality of cohomology.
• Lecture 16:Differential forms with distribution coefficients. Serre duality.
• Lecture 17:Proof of Serre duality. Dolbeault resolutions for forms with distribution coefficients. Baby version of elliptic regularity.
• Lecture 18: Line bundles and H^1(X, O^\star). Chern classes.
• Lecture 19: More on line bundles and Chern classes. Chern forms. 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. I showed that any line bundle is the line bundle of a divisor. Any meromorphic function has as many zeros as poles.
• Lecture 21: Riemann-Roch. I showed how to find the cohomology of line bundles over a genus g surface provided that the degrees are less or equal to 0 or higher or equal to 2g-2. I showed the degree of K is 2g-2. The genus is a topological invariant. Genus 0 curves are isomorphic to P^1.
• Lecture 22: Applications of Riemann-Roch. Genus 1 curves are isomorphic to their Picard. Abel-Jacobi maps. Clifford's theorem.
• Lecture 23: Linear series. Basepointfree, ample and very ample line bundles/linear series. In degree at least 2g, we get basepointfreeness. In degree 2g+1 or more we get very ampleness. K_X is basepointfree in genus at least 1. Canonical morphisms.
• Lecture 24: Genus 1 curves can be viewed as cubics in P^2 or intersection of two quadrics in P^3. Hyperlliptic curves.
• Lecture 25: Genus 2 curves are hyperelliptic. K_X is very ample if X is not hyperelliptic. 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 26: Moduli space of genus g curves. Fine vs. coarse representability. Explained the Mumford-Harris-Eisenbud theorem.