## Math 625, Riemann Surfaces

## Fall 2017

**Instructor:** Richard Hain
**Text:** R. Miranda: *Algebraic Curves and Riemann Surfaces*, Graduate
Studies in Mathematics, Vol 5, American Mathematical Society, 1995.

**References:**

- C. H. Clemens:
*Herbert A scrapbook of complex curve theory*
(Second edition), Graduate Studies in Mathematics, 55, American
Mathematical Society, Providence, RI, 2003.
- M. Cornalba:
*The theorems of Riemann-Roch and Abel*, Lectures
on Riemann surfaces (Trieste, 1987), 302-349, World Sci. Publ., 1989.
(pdf)
- Donaldson, Simon:
*Riemann Surfaces*, Oxford University Press,
2011.
- H. Farkas, I. Kra:
*Riemann Surfaces*, Graduate Texts
in Mathematics, 71. Springer-Verlag, New York, 1980.
- Forster, Otto:
* Lectures on Riemann surfaces*, Graduate
Texts in Mathematics, 81. Springer-Verlag, New York, 1991.
- Phillip Griffiths:
*Introduction to algebraic curves*,
Translations of Mathematical Monographs, 76. American Mathematical
Society, Providence, RI, 1989.
- Phillip Griffiths, Joseph Harris:
*Principles of algebraic
geometry*, Wiley Classics. (ff. Chapter 2).
- R. C. Gunning:
*Lectures on Riemann surfaces*, Princeton
Mathematical Notes Princeton University Press, Princeton, N.J. 1966.
- David Mumford:
* Curves and their Jacobians*, The University
of Michigan Press, Ann Arbor, Mich., 1975.

The most basic of these are the books by Forster, Farkas and Kra, Griffiths
(his China lectures) and Gunning. Mumford's lectures have been reprinted as
part of his *Red Book of Varieties and Schemes*.
**Topics:**

- Introduction
- definitions (Riemann surfaces, holomorphic and meromorpic maps)
- examples (algebraic curves, tori, translation surfaces, etc)
- local structure of holomorphic maps, branched coverings, etc.
- Galois theory of meromorphic functions
- Riemann Hurwicz formula and Plucker's formula

- Complex calculus
- in one complex variable
- basics of several complex variables
- complex manifolds
- holomorphic implicit function theorem

- Compact Riemann Surfaces
- divisors and line bundles
- Bezout's Theorem
- Riemann-Roch Theorem
- Serre Duality
- applications
- elliptic curves
- hyperelliptic curves
- linear systems and maps to projective space
- the jacobian
- Abel's Theorem and Jacobi inversion

**Assignments:**

- Assignment 1: (pdf) (Due: Thursday, September 14)
- Assignment 2: (pdf) (Due: Thursday, September 26)
- Assignment 3: (pdf) (Due: Thursday, October 12)
- Assignment 4: (pdf) (Due: Tuesday, October 31)
- Assignment 5: (pdf) (Due: Thursday, November 16)
- Assignment 5: (pdf) (Due: Thursday, December 14)

**Notes:**

Notes on the local structure of proper maps (pdf)

Return to: Richard Hain *
Duke Mathematics Department *
Duke University