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）资助的。本文的主要目的是寻找极化K3曲面模空间的一种几何意义的紧化，作为Mumford的环面紧化理论的一个实例，该理论要求在某个锥上具有扇形结构（关于某个离散群的等变），而激励性的观察是，来自Mori理论和镜像对称的基本思想产生了一个典型的这样的风扇：也就是说，锥的问题原来是一个自然的模空间的总空间中的有效因子的锥，Dolgachev的镜子极化K3的模量，和森风扇是一个典型的风扇结构上的任何品种的有效锥。希望是（在其边界附近）相关联的复曲面品种进行了一个典型的普遍家庭的对K3表面与因子。这样一个家庭的存在自然会推广镜像对称计划的Kontsevich-Soibelman和Gross-Siebert，并在同一时间统一这项工作与瑟斯顿的工作三角形的2-sphere，和Looijenga的猜想smoothings的尖点奇点。根据量子场论的物理宇宙是由一个所谓的Calabi-Yau流形-一种特定的几何对象。因此，重要的是要理解这样的对象的集合-所谓的卡-丘流形的模空间，特别是要理解这个空间在无穷远处的行为，即在其边界附近，或者等价地，理解卡-丘流形如何退化。有一个案例很好理解：一个复杂的一维卡-丘流形是一个甜甜圈的表面，当我们移动到模空间的边界时，甜甜圈上的一个环会收缩，模空间边界上的极限点对应于将这个环挤压成一个点--这就产生了一个空间，你可以通过取一个真实的二维球体，并将其弯曲，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，复杂的维度2中发生了什么，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.