SPECIAL TOPIC COURSE RIEMANN SURFACES
VSEVOLOD SHEVCHISHIN
Abstract. The course is devoted to the theory of Riemann surfaces and algebraic curves. We shall cover main results and topics such as Riemann-Hurwitz formula, elliptic curves and Weier- strass' ℘-function, divisors and line bundles, Jacobian variety of an algebraic curve, canonical divisor of an algebraic curve, holomorphic vector bundles, Czech- and Dolbeault-cohomology, Riemann-Roch formula. 1. Generalities This course is devoted to the theory of Riemann surfaces (RS) and algebraic curves. The foundation of the theory was made in XIX century by Bernhard Riemann, Niels Henrik Abel, Carl Gustav Jacobi, Karl Weierstrass, Adolf Hurwitz and others. Nowadays Riemann surfaces play a remarkable role in modern mathematics and appear in various parts such as algebraic geometry, number theory, topology, they also are very important in modern mathematical physics. The theory of Riemann surfaces includes and combines methods coming from three elds of mathematics: analysis, topology, and algebra (and algebraic geometry). One of the goals of the course is to show the interaction and mutual in uence of ideas and approaches.
2. Prerequisites. • Basic course in complex analysis (Theory of functions of one complex variable). • Basic facts from dierential geometry and algebraic topology.
3. Tentative syllabus (1) Riemann surfaces and holomorphic maps. Meromorphic functions. (2) Structure of holomorphic maps. Riemann-Hurwitz formula. (3) Conformal and quasiconformal maps. (4) Dierential forms on RS, Cauchy-Riemann operator, Cauchy-Green formula and solution of ∂¯-equation. (5) Divisors and holomorphic line bundles on RS-s. (6) Holomorphic vector bundles on RS-s and their cohomology. Serre duality. (7) Canonical line bundle and Riemann-Roch formula. (8) Complex curves in projective spaces. Planar curve and Plucker formula. (9) Elliptic curves, ℘- and Θ-functions. (10) Jacobians of algebraic curves. (11) Hyperelliptic curves and Weierstrass points.
4. References. The main reference is the book of Farkas and Kra [1]. References [24] can be used as comple- mentary. The book of Gri ths-Harris [5] is a must-read for all mathematicians. [1] Hershel M. Farkas, Irwin Kra; Riemann Surfaces. xvi+363pp; 1992; Graduate Texts in Mathematics, 71, Springer-Verlag. [2] Rick Miranda; Algebraic Curves and Riemann Surfaces. xxi+390pp; 1995; Graduate Stud- ies in Mathematics, 5, AMS, Providence, RI. [3] Otto Forster; Lectures on Riemann surfaces. viii+254pp; 1981 Graduate Texts in Mathe- matics, 81; Springer-Verlag. 1 2 RIEMANN SURFACES
[4] Martin Schlichenmaier; An introduction to Riemann surfaces, algebraic curves and moduli spaces. 2nd ed.; xiii+217pp; 2007; Springer-Verlag. [5] Phillip Gri ths, Joseph Harris; Principles of algebraic geometry. 2nd ed.; xii+813 pp; 1994; John Wiley & Sons, NY.
Department of mathematics, National Research University Higher School of Economics, Vav- ilova 7, Moscow 117312, Russia, Tel.: +7 (495) 772-95-90 *4144, *4147 E-mail address: [email protected]