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。提出的研究的主要目的是找到偏振K3表面模空间的几何上有意义的紧化，作为Mumford的环面紧化理论的一个实例。理论需要一个风扇结构在某锥(等变化对某离散组),观察和激励是来自Mori理论的基本思想和镜面对称生产规范这样的粉丝:即问题是锥的锥有效因子的总空间的自然模空间,Dolgachev极化k3的镜子模,Mori扇是一种规范风扇结构的有效锥任何品种。我们的希望是（在它的边界附近）相关的环变携带一个带除数的K3曲面对的正则泛族。这样一个族的存在将很自然地推广kontsevic - soibelman和Gross-Siebert的镜像对称规划，同时将这一工作与Thurston关于2球三角剖分的工作以及Looijenga关于顶点奇点光滑的猜想统一起来。根据量子场论，物理宇宙是由所谓的卡拉比-丘流形控制的——一种特定的几何物体。因此，理解这些对象的集合——所谓的Calabi-Yau流形的模空间是很重要的，特别是要理解这个空间在无穷远处的行为，即在它的边界附近，或者等价地，理解Calabi-Yau流形是如何退化的。一种情况很好理解：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，甜甜圈上的一个环会收缩，模空间边界上的极限点对应于将这个环压成一个点——这就产生了你通过取一个实二维球体并弯曲它直到两点相互接触而获得的空间。该项目的目标是了解下一个更高维度，即复杂维度2中发生的情况，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.