Content of the course
Abelian varieties are of central importance in Algebraic Geometry, Complex Geometry and Number Theory. On the one hand, they are amenable to explicit computation; on the other hand, they arise naturally in many different areas of mathemtics, for example, in the study of line bundles on algebraic varieties or in the study of (abelian) extensions of certain number fields. In this lecture we would like to give an introduction to the theory of (complex) abelian varieties.
After a brief discussion of 1-dimensional abelian varieties (i.e. elliptic curves), we will turn our attention to higher dimensional complex tori. We will discuss line bundles on complex tori and describe their sections explicitly in terms of theta functions. The question of the existence of projective embeddings leads naturally to the definition of complex abelian varieties. If time permits, we will discuss the Hodge decomposition of complex tori, compute the cohomology of line bundles on complex tori and discuss moduli spaces of polarized abelian varieties.
Formally, this is a continuation of the lecture Complex Geometry I, but we try to make the lecture as self-contained as possible. Please, do not hesitate to contact me if you have any questions.
For more details, please enroll in the Moodle course (the password is the name of a mathematician in the title of the lecture, 4 letters).