Birational geometry, symplectic varieties, and moduli spaces

这个项目解决了代数几何中的三个中心问题：一个极化全纯辛簇的取样锥可以从它的Hodge结构中计算出来吗？什么是高维簇的紧模空间的正确的函数式定义(它们可能有什么用处)？模空间的二次几何在多大程度上支配着相关几何不变量理论问题的行为，反之亦然？这些问题以复杂而美丽的方式交织在一起：交集理论构造支配着稳定曲线和全纯辛簇的模空间上的曲线类。K3曲面的Torelli定理是其模理论的起点；高维全纯辛流形缺乏这一结果是分析其充分锥的主要动力。K3曲面的模空间的几何紧致这一难以捉摸的梦想推动了几何不变理论和模空间之间的相互作用。代数几何是研究由多项式方程定义的几何对象的学科，称为簇。变种的例子包括圆、椭圆、抛物线、球面等。一个基本的问题是对给定类型的所有变种进行分类。一种方法是分析由给定次数的多项式定义的所有变元，例如高中解析几何中学习的二次曲线。在这里，变种的类型用代数术语表示。或者，人们可以研究具有共同几何特征的所有变种，例如，具有给定数值不变量的变种。这个项目解决了代数量和几何量之间的相互作用，以及它们如何管理变种家族的行为。

This project addresses three central problems in algebraic geometry:Can one compute the ample cone of a polarized holomorphic-symplectic variety from its Hodge structure? What is the right functorial definition for compact moduli spaces of higher-dimensional varieties (and what might they be good for)? To what extent does the birational geometry of moduli spaces govern the behavior of related Geometric Invariant Theory problems, and vice versa? These questions are intertwined in intricate and beautiful ways: Intersection-theoretic constructions govern curve classes on both the moduli space of stable curves and holomorphic-symplectic varieties. The Torelli Theorem for K3 surfaces is the starting point for their moduli theory; the lack of such a result for higher dimensional holomorphic-symplectic manifolds is a major impetus for analyzing their ample cones. The elusive dream of a geometric compactification for the moduli space of K3 surfaces animates work on the interplay between Geometric Invariant Theory and moduli spaces.Algebraic geometry is the study of geometric objects defined by polynomial equations, which are called varieties. Examples of varieties include circles, ellipses, parabolas, spheres, etc. A fundamental problem is to classify all the varieties of a given type. One approach is to analyze all the varieties defined by polynomials of given degree, e.g., the conic sections studied in high school analytic geometry. Here the type of the variety is expressed in algebraic terms. Alternately, one can study all the varieties sharing common geometric characteristics, e.g., those with given numerical invariants. This project addresses the interplay between the algebraic and geometric quantities, and how these govern the behavior of families of varieties.