Geometry of abelian varieties and their moduli

PI计划研究主极化复阿贝尔变体的模空间几何。他建议（与合作者一起）研究模空间上的锥有效因子，因子的相交理论，以及由向量值siegel模形式引起的紧化。PI进一步提出检验Kummer变量的几何，特别是它们的割线，推广和扩展与三割线猜想相关的早期工作，以定义模空间的自然分层。为此，他还提出研究2Theta线性系统的几何，旨在得到基轨迹和奇点的描述，将Riemann的理论推广到Theta。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这项研究。阿贝尔变分及其模空间是代数几何和数论中的经典中心对象之一。阿贝尔变种是具有群结构的代数变种，即本质上是定义两点和运算的几何对象。复杂维度的一种情况-椭圆曲线，也可以被认为是三次曲线，如^2=x(x-1)(x-a) -产生了许多美丽的经典几何结构：例如，为了计算两点的和，在这两点之间画一条线，取它与图形的第三个交点，并改变y的符号。在每个维度中都有许多不同的阿贝尔变体（改变上面的“a”给出不同的对象）。理解模空间的几何——给定维度的所有可能的阿贝尔变体的集合——可以产生对代数几何、数论、可积系统和弦理论等各种主题的见解。

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.