Complex geometry

The lecture stretches on the whole academic year 2016/17 and there will be one course a week. There will be also exercise sessions each two weeks. The summer semester course is mostly independent from the previous winter semester part. The summer semester is devoted to the introduction to the topics in complex geometry of Riemann surfaces and holomorphic line bundles.
We will cover:

- Holomorphic line bundles on Riemann surfaces,
- Divisors,
- Jacobean variety,
- Riemann-Roch theorem,
- Bergman kernel on Riemann surfaces,
- Abel theorem, Jacobi inversion theorem,
- Riemann's theta functions,
- Prime form,
- Bosonisation formulas on Riemann surfaces,
- Quantum Hall states on Riemann surfaces,
- Further topics.

The course is structured with the goal to give all the necessary ingredients for proving Fay's bosonisation formulas on Riemann surfaces. The tools that we will learn in this course are also relevant for topics in mathematical physics, such as conformal field theory, string theory and Quantum Hall effect.

Lectures: 5, 6, 7, 8, 9, 12, 13, 14

Homeworks: 7, 8, 9

Literature:

[1] J. Jost, "Compact Riemann surfaces: An Introduction to Contemporary Mathematics", Springer, 1997.
[2] X. Ma, G. Marinescu "Holomorphic Morse inequalities and Bergman kernels", Progress in Mathematics v. 254, Birkhauser, 2007.
[3] J. Fay, "Theta functions on Riemann surfaces", Lecture Notes in Mathematics, v. 352, Springer-Verlag, 1973.
[4] H. M. Farkas, I. Kra , "Riemann surfaces", Graduate Texts in Mathematics Volume 71, Springer, 1992.
[5] D. Mumford, "Tata lectures on Theta, I ", Birkhauser, 1983.
[6] A. Bobenko, "Compact Riemann surfaces", lecture notes.
[7] L. Alvarez-Gaume, J.-B. Bost, G. Moore, P. Nelson, C. Vafa, "Bosonization on Higher Genus Riemann Surfaces", Commun. Math. Phys. 112 (1987) 503-552.
[8] S. Klevtsov, "Geometry and Large N limits in Laughlin states", Travaux Math., v. XXIV, (2016) 63-127, arXiv:1608.02928.