Birational Geometry of Moduli Spaces

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。该研究的主要目的是找到极化K3曲面的模空间的几何意义紧化，该紧化是作为Mumford环面紧化理论的一个实例而得到的。该理论要求在某个锥体上有一个扇形结构(关于某个离散群是等变的)，而激励观察是来自Mori理论和镜像对称性的基本思想产生了一个典型的这样的扇体：即所讨论的锥体是自然模空间全空间中的有效因子的锥体，Dolgachev的镜像到极化K3s的模上，Mori扇体是任何种类的有效锥体上的典范扇形结构。人们希望(在其边界附近)相关的环面簇具有一个标准普适族，该族由成对的带因子的K3曲面组成。这一族的存在自然地推广了Kontsevich-Soibelman和Gross-Siebert的镜像对称程序，同时将这一工作与瑟斯顿关于2-球面三角剖分的工作以及Looijenga关于尖点奇点光滑的猜想结合起来。根据量子场论，物理宇宙由所谓的Calabi-Yau流形--一种几何对象--控制。因此，重要的是要了解这类对象的集合--Calabi-Yau流形的所谓的模空间，特别是要了解该空间在无穷远处(即在其边界附近)的行为，或者等价地理解Calabi-Yau流形如何退化。一种情况很容易理解：一个复杂的一维Calabi-Yau流形是一个甜甜圈的表面，当一个人移动到模空间的边界时，甜甜圈上的一个环收缩，模空间边界上的极限点对应于将这个环压碎成一个点--这产生了一个空间，通过取一个真实的二维球体并弯曲它，直到两个点相互接触。该项目的目标是了解在下一个更高的维度，即复维度2中发生的事情，从而描述所谓的K3曲面的模空间的边界行为。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The main objective of the proposed research is to find a geometrically meaningful compactification of the moduli space of polarized K3 surfaces, obtained as an instance of Mumford's theory of toroidal compactifications.The theory requires a fan structure on a certain cone (equivariant with respect to a certain discrete group), and the motivating observation is that elementary ideas from Mori theory and mirror symmetry produce a canonical such fan: Namely the cone in question turns out to be the cone of effective divisors in the total space of a natural moduli space, Dolgachev's mirror to moduli of polarized K3s, and the Mori fan is a canonical fan structure on the effective cone of any variety. The hope is that (near its boundary) the associated toric variety carries a canonical universal family of pairs of K3 surface with divisor. The existence of such a family would naturally generalize the mirror symmetry programs of Kontsevich-Soibelman and Gross-Siebert, and at the same time unifying this work with Thurston's work on triangulations of the 2-sphere, and Looijenga's conjecture on smoothings of cusp singularities.According to quantum field theory the physical universe is controlled by a so called Calabi-Yau manifold -- a certain kind of geometric object. As a result it is important to understand the set of such objects -- the so called Moduli space of Calabi-Yau manifolds, and in particular to understand the behavior of this space at infinity, i.e. near its boundary, or equivalently, to understand how Calabi-Yau manifolds can degenerate. One case is well understood: A complex 1-dimensional Calabi-Yau manifold is the surface of a doughnut, and as one moves to the boundary of the moduli space, one loop on the doughnut shrinks, the limiting point on the boundary of the moduli space corresponds to crushing this loop to a point -- which yields the space you obtain by taking a real 2 dimensional sphere and bending it until two points touch each other. The goal of the project is to understand what happens in the next higher dimension, complex dimension two, and thus to describe the boundary behavior of the moduli space of so called K3 surfaces.