Math 272, Riemann Surfaces

Fall 2006

Topics:

• Introduction
  • definitions
  • examples
• Basic Properties
  • branched coverings
  • Galois theory of proper holomorphic maps
  • Riemann-Hurwicz formula (I)
  • sheaves and analytic continuation
  • algebraic functions
• Differential Forms and Cohomology
  • integration formulas
  • meromorphic differentials, residues
  • harmonic differentials
  • Weyl's Lemma
  • Dolbeault cohomology
  • Hodge theorem and finiteness
  • holomorphic line bundles
  • sheaf cohomology
  • Serre duality
• Compact Riemann Surfaces
  • divisors and line bundles
  • Riemann-Roch Theorem
  • applications
  • elliptic curves
  • hyperelliptic curves
  • linear systems and maps to projective space
  • the jacobian
  • Abel's Theorem and Jacobi Inversion
• Topics (time permitting) selected from:
  • Weierstrass points, automorphisms
  • moduli of curves of low genus
  • theta functions and modular forms

References:

1. C. H. Clemens: Herbert A scrapbook of complex curve theory (Second edition), Graduate Studies in Mathematics, 55, American Mathematical Society, Providence, RI, 2003.
2. H. Farkas, I. Kra: Riemann Surfaces, Graduate Texts in Mathematics, 71. Springer-Verlag, New York, 1980.
3. Phillip Griffiths: Introduction to algebraic curves, Translations of Mathematical Monographs, 76. American Mathematical Society, Providence, RI, 1989.
4. Phillip Griffiths, Joseph Harris: Principles of algebraic geometry, Wiley Classics. (ff. Chapter 2).
5. R. C. Gunning: Lectures on Riemann surfaces, Princeton Mathematical Notes Princeton University Press, Princeton, N.J. 1966.
6. R. Miranda: Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics, Vol 5, American Mathematical Society, 1995.
7. David Mumford: Curves and their Jacobians, The University of Michigan Press, Ann Arbor, Mich., 1975.
8. Forster, Otto: Lectures on Riemann surfaces, Graduate Texts in Mathematics, 81. Springer-Verlag, New York, 1991.

Assignments:

• Assignment 1 (pdf)

Old Assignments:

• Assignment 1 (pdf)
• Assignment 2 (pdf)
• Assignment 3 (pdf)
• Assignment 4 (pdf)