Geometry of abelian varieties and their moduli

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The PI is proposing to investigate geometry of the moduli space ofprincipally polarized complex abelian varieties. He proposes toundertake (together with collaborators) a study of the cone ofeffective divisors on the moduli space, of the intersection theoryof divisors, and of the compactifications induced by vector-valuedSiegel modular forms. The PI further proposes to examine the geometryof Kummer varieties, and in particular of their secants, generalizingand extending the earlier work related to the trisecant conjecture, todefine a natural stratification of the moduli space. To this end, healso proposes to investigate the geometry of the 2Theta linearsystem, aiming to get a description of base loci and singularitiesgeneralizing Riemann's theory for Theta. The PI proposes to combinemethods of algebraic geometry, number theory/modular forms, and evenintegrable systems to undertake this study.Abelian varieties and their moduli spaces are one of the classicalcentral objects in algebraic geometry and number theory. An abelianvariety is an algebraic variety with a group structure, i.e.essentially it is a geometric object on which the operation of takinga sum of two points is defined. The complex dimension one case -elliptic curves, which can also be thought of as cubic curves likey^2=x(x-1)(x-a) - gives rise to many beautiful classical geometricconstructions: for example, to compute the sum of two points one drawsthe line through these two points, take its third point of intersectionwith the graph and change the sign of y. In every dimension there aremany different abelian varieties (varying "a" above gives differentobjects). Understanding the geometry of the moduli space - the set ofall possible abelian varieties of a given dimension - couldyield insights in topics as varied as algebraic geometry, numbertheory, integrable systems, and string theory.