On the Motivic Goettsche Invariants

关于 Motivic Goettsche 不变量

• 批准号: 1503621
• 负责人: Yu-jong Tzeng
• 金额: $ 14.3万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1503621&HistoricalAwards=false
• 关键词: Motivic; Goettsche; Invariants

Algebraic geometry is the study of solutions of polynomial equations, with many applications in other parts of mathematics, physics, and engineering. Some of the well-studied objects in algebraic geometry are the projective plane (the plane with additional "points at infinity") and curves on it. A natural question is that of finding the number of curves that satisfy certain special conditions, such as passing through fixed points or having special shapes, which was the starting point of a branch of algebraic geometry called enumerative geometry. During the past twenty years, the interaction between enumerative geometry and string theory in physics has resulted in many exciting new developments and has been a central area of current research. This research project aims to define and study new invariants, in a broader sense on general algebraic surfaces, that have connections to both mathematics and physics. These invariants contain sophisticated information about the number of singular curves, the geometric properties of the surface, and the parametrization space of singular curves. The project envisages development of a theoretical framework and tools for computation, as well as establishing connections between different subjects in mathematics. It will add to the understanding of geometry of algebraic varieties and has application to the description of spacetime in string theory.This research project treats several problems in algebraic geometry, especially about the enumeration of singular curves on algebraic surfaces and its motivic generalization. Previous work has established that the numbers of varieties with certain given singularities satisfy universal formulas. This project concerns the generalization of these universal formulas in the motivic sense. The research defines a sequence of motivic invariants and studies their quantitative properties as well as their geometric interpretation. The work aims to extend the existing connections with Gromov-Witten theory, stable pairs theory, and modular forms for these new invariants, and to compare them with other motivic curve-counting invariants. One of the advantages of these new invariants is that they satisfy nice formulas and only depend on topological intersection numbers.

代数几何是研究多项式方程解的学科，在数学、物理和工程的其他领域有许多应用。在代数几何中，一些研究得很好的对象是射影平面（有额外的“无穷点”的平面）和曲线。一个自然的问题是找到满足某些特殊条件的曲线的数量，例如通过固定点或具有特殊形状，这是代数几何的一个分支称为枚举几何的起点。在过去的二十年中，数列几何和弦理论在物理学中的相互作用产生了许多令人兴奋的新发展，并已成为当前研究的一个中心领域。这个研究项目的目的是定义和研究新的不变量，在更广泛的意义上的一般代数曲面，有联系的数学和物理。这些不变量包含奇异曲线的数量、曲面的几何性质和奇异曲线的参数化空间等复杂信息。该项目设想开发计算的理论框架和工具，并建立数学中不同学科之间的联系。它将增加对代数变量几何的理解，并应用于弦理论中时空的描述。本课题研究代数几何中的几个问题，特别是代数曲面上奇异曲线的枚举及其动机推广。以前的工作已经证实，具有某些给定奇点的变异数满足全称公式。这个项目涉及这些普遍公式在动机意义上的推广。本研究定义了一系列动力不变量，并研究了它们的数量性质及其几何解释。本文的目的是扩展这些新不变量与Gromov-Witten理论、稳定对理论和模形式的现有联系，并将它们与其他动机计数曲线不变量进行比较。这些新的不变量的优点之一是它们满足很好的公式，并且只依赖于拓扑交数。