From TorontoMathWiki

Riemann Surfaces

Instructor: Marco Gualtieri

Class schedule: Fridays 2-5 pm in BA6180, starting Friday 13th January, 2012

Outline

These may be re-arranged, but it gives an idea of what I'd like to explore in this class.

1. Intrinsic structure of a Riemann surface. Line bundles+divisors, sheaves, covering spaces, elliptic curves, Riemann-Roch
2. Embedding the surface into projective space. Linear systems, plane algebraic curves, classical projective geometry
3. Moduli spaces of curves, GIT, Deligne-Mumford
4. Vector bundles, Narasimhan-Seshadri, moduli of vector bundles
5. Derived category of coherent sheaves, stability conditions

I'd also like to touch on Strebel differentials

The course will not dwell too much on Hodge theory, uniformization or other analytic issues, for obvious reasons given the list of topics above.

Student work

This will be essentially a reading course + seminar for students. Evaluation will be based on participation (depending on enrollment, I will ask people to present topics and provide lecture notes) as well as a final project based on giving a modern account of some classical topic/paper. More on this after the first class.

References

1. Basic theory
   1. Gunning Riemann surfaces
   2. Forster's text on Riemann Surfaces
   3. Griffiths intro to alg. curves
   4. Griffiths & Harris
   5. Arbarello, Cornalba, Griffiths & Harris algebraic curves I and II
   6. Hitchin on Riemann surfaces and integrable systems
   7. Fulton's Algebraic curves
   8. Kirwan on algebraic curves
   9. Walker's Algebraic curves
   10. Schlichenmaier on Riemann surfaces and mathematical physics
   11. Jost on compact Riemann surfaces
2. Plane curves
   1. Brieskorn-Knorrer Plane algebraic curves
3. Moduli of curves
   1. Harris & Morrison moduli of curves
   2. Gieseker's moduli of curves
4. Integrable systems
   1. Beauville on Jacobians of spectral curves
   2. Donagi & Markman on spectral covers
   3. Hitchin on monopoles and minimal surfaces
   4. Quillen on the determinant line bundle
   5. Markman spectral curves
   6. Polishchuk on Abelian varieties
5. Sheaves
   1. Grauert-Remmert coherent analytic sheaves
   2. Hartshorne
6. Moduli of Vector Bundles
   1. Gunning vector bundles
   2. Atiyah's thesis
   3. Potier
   4. Alvarez-Consul and King
   5. Garcia-Prada appendix to Wells' text

Project topics

1. The Riemann-Roch theorem
2. The theorem of Narasimhan-Seshadri, moduli of vector bundles
3. Classification of vector bundles on the elliptic curve, thesis of Atiyah
4. Intrinsic geometry of singular curves
5. Strebel differentials and the recent work of Gaiotto-Moore-Neitzke
6. Super Riemann surfaces
7. Fat points, fat curves
8. Elliptic surfaces, the Kodaira classification
9. Elliptic normal curves, papers of Hulek et al.
10. Moduli of pairs, papers of Bradlow, Thaddeus, and Polishchuk
11. Determinant bundles
12. Fundamental group of complement of plane curve
13. Orbifolds and Riemann surfaces
14. Vertex algebras
15. Integrable systems:
    1. Spinning tops
    2. Lax formalism
    3. Hitchin system
    4. Spectral curves, papers of Beauville and Hitchin
    5. Discrete integrable systems and Riemann surfaces
16. b-complex curves