Complex Geometry

Accademic year 2018-2019 (6 cfu, 42 hours)
Frist part: Prof. E. Colombo, Second part: Prof. B. van Geemen.

Prerequisites:
Basics of complex analytic functions, differentiable and complex manifolds (for complex analysis these notes might be useful).

Program:
Topology and de Rham cohomology of Riemann suraces [Fu].

Holomorphic maps between compact Riemann surfaces. The Riemann-Hurwitz theorem. Compact Riemann surfaces and algebraic curves.

Elliptic curves: complex tori, their moduli space and, time permitting, the j-invariant and modular forms [S].

The Riemann-Roch theorem and its application to projective models, the canonical model [Mi].

A brief introduction to the Jacobian of a Riemann surface, the Picard group, theta functions and the theorems of Abel and Riemann [GH].


Contents of the lectures.

Exercises (Algant students should consign between one and three exercises)
for the first part: Exercises A.pdf.
for the second part: Exercises B.pdf.

Bibliography:
[D] S. Donaldson, Riemann Surfaces, Oxford Graduate Texts in Mathematics 22, Oxford, 2011.
[Fo] O. Forster, Lectures on Riemann Surfaces, GTM 81, Springer, New York, 1981.
[Fu] W. Fulton, Algebraic Topology. A First Course, GTM 153, Springer, New York, 1995.
[GH] Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, John Wiley and Sons 1978.
[Mi] R. Miranda, Algebraic Curves and Riemann Surfaces. American Mathematical Society 1995.
[Na] M. Namba, Geometry of Projective algebraic Curves, Marcel Dekker, Inc. 1984.
[S] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, GTM 151. Springer-Verlag 1994.