The PI is proposing to investigate geometry of the moduli space of principally polarized complex abelian varieties. He proposes to undertake (together with collaborators) a study of the cone of effective divisors on the moduli space, of the intersection theory of divisors, and of the compactifications induced by vector-valued Siegel modular forms. The PI further proposes to examine the geometry of Kummer varieties, and in particular of their secants, generalizing and extending the earlier work related to the trisecant conjecture, to define a natural stratification of the moduli space. To this end, he also proposes to investigate the geometry of the 2Theta linear system, aiming to get a description of base loci and singularities generalizing Riemann's theory for Theta. The PI proposes to combine methods of algebraic geometry, number theory/modular forms, and even integrable systems to undertake this study.
Abelian varieties and their moduli spaces are one of the classical central objects in algebraic geometry and number theory. An abelian variety is an algebraic variety with a group structure, i.e. essentially it is a geometric object on which the operation of taking a sum of two points is defined. The complex dimension one case - elliptic curves, which can also be thought of as cubic curves like y^2=x(x-1)(x-a) - gives rise to many beautiful classical geometric constructions: for example, to compute the sum of two points one draws the line through these two points, take its third point of intersection with the graph and change the sign of y. In every dimension there are many different abelian varieties (varying "a" above gives different objects). Understanding the geometry of the moduli space - the set of all possible abelian varieties of a given dimension - could yield insights in topics as varied as algebraic geometry, number theory, integrable systems, and string theory.
