Geometry of abelian varieties and their moduli

PI建议研究主要极化复阿贝尔簇的模空间的几何。他提出了一个研究的锥ofeffective因子的模空间，交叉theoryof因子，和compactifications诱导vector-valuedSiegel模形式。PI进一步建议检查几何的库默品种，特别是他们的割线，generalizingand扩展早期的工作有关的trisecant猜想，以定义一个自然分层的模空间。为此，他还建议调查的几何2 θ linearsystem，旨在得到一个描述的基本轨迹和奇异性推广黎曼理论θ。PI建议将代数几何、数论/模形式甚至可积系统的方法结合起来进行研究。阿贝尔簇及其模空间是代数几何和数论的经典中心对象之一。阿贝尔簇是具有群结构的代数簇，即它本质上是定义了两点求和运算的几何对象。复杂的一维情形--椭圆曲线，也可以被认为是像^2 =x（x-1）（x-a）这样的三次曲线--产生了许多美丽的经典几何构造：例如，为了计算两点的和，通过这两点画一条直线，取它与图的第三个交点并改变y的符号。在每一个维度上都有许多不同的阿贝尔变体（改变上面的“a”会得到不同的阿贝尔）。理解模空间的几何--给定维度的所有可能的阿贝尔簇的集合--可以在代数几何、数论、可积系统和弦理论等不同的主题中产生深刻的见解。

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.