Birational geometry, symplectic varieties, and moduli spaces

双有理几何、辛簇和模空间

• 批准号: 0901645
• 负责人: Brendan Hassett
• 金额: $ 42.8万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901645&HistoricalAwards=false
• 关键词: Birational; geometry; symplectic; varieties; moduli

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.

该项目解决了代数几何中的三个核心问题：能否根据霍奇结构计算极化全纯辛簇的大锥体？ 高维簇的紧模空间的正确函数定义是什么（以及它们有什么用处）？ 模空间的双有理几何在多大程度上控制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：交集理论构造控制着稳定曲线和全纯辛簇的模空间上的曲线类。 K3 曲面的托雷利定理是其模量理论的起点； 高维全纯辛流形缺乏这样的结果是分析其充足锥体的主要动力。 K3 曲面模空间的几何紧化这一难以捉摸的梦想激发了几何不变量理论和模空间之间相互作用的研究。代数几何是对由多项式方程（称为簇）定义的几何对象的研究。 变体的例子包括圆形、椭圆形、抛物线、球体等。一个基本问题是对给定类型的所有变体进行分类。 一种方法是分析由给定次数的多项式定义的所有类型，例如高中解析几何中研究的圆锥曲线。 这里，品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体，例如具有给定数值不变量的变体。 该项目解决了代数和几何量之间的相互作用，以及它们如何控制品种家族的行为。