Riemann Surfaces

Riemann surfaces are nothing but one-dimensional complex manifolds; as such, they provide the first nontrival example both to the general theory of complex manifols and to the theory of algebraic curves in algebraic geometry. In this course we concentrate on geometric rather than analytic aspects of the theory.

Prerequisites

Complex Analysis (with analytic continuation), Topology I (with coverings).

Contents of the course

  1. Riemann surfaces: definition, simple examples, holomorphic and meromorphic functions.
  2. Compact Riemann surface corresponding to an algebraic equation. Riemann’s existence theorem (without proof).
  3. Genus of a compact Riemann surface. Ramified coverings. Sum of orders of a meromorphic function. Riemann–Hurwitz formula.
  4. Holomorphic and meromorphic differentials. Sum of orders of a meromorphic differential.
  5. Residues. Sum of residues of a meromorphic differential on a compact Riemann surface.
  6. Divisors. Linear systems.
  7. Mittag–Leffler problem on compact Riemann surfaces. Upper bound on the dimension of a linear system via residues. Riemann–Roch theorem (statement).
  8. Poincaré residue. Construction of holomorphic and meromorphic differentials on smooth plane curves and smooth complete intersections in projective space.
  9. Canonical linear system and canonical mapping. Characterization of hyperelliptic Riemann surfaces.
  10. Line bundles on complex manifolds and, in particular, on Riemann surfaces. Line bundles and divisors. Line bundles and linear systems.
  11. Inflection points of linear systems. Weierstrass points.
  12. Proof of Riemann–Roch theorem using adèles (after A.Weil).
  13. Jacobian variety. Abel’s and Jacobi’s theorems. Jacobi’s inversion theorem.

Grading policy

The numerical grade equals 0.3*(homework)+0.3*(midterm exam)+0.4*(final exam). The midterm and final exams are written, open-book.

Textbooks

There is no single textbook which the course follows. The following texts, among others, may be of use to the student.

  • R.C.Gunning. Lectures on Riemann Surfaces. Princeton University Press: Princeton, NJ, 1966.
  • S.Lang. Introduction to Algebraic and Abelian Functions. Springer-Verlag, 1982.
  • R.Miranda. Algebraic Curves and Riemann Surfaces. Amer. Math. Soc., Providence, RI, 1995.
  • R.Narasimhan. Compact Riemann Surfaces. Birkhäuser, 1992.
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