A-infinity structures and derived categories in algebraic geometry

代数几何中的 A-无穷大结构和派生范畴

• 批准号: 1400390
• 负责人: Alexander Polishchuk
• 金额: $ 15.5万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1400390&HistoricalAwards=false
• 关键词: infinity; structures; derived; categories; algebraic; infinity; structures; derived; categories; algebraic

This research project is in the field of algebraic geometry with some connections to string theory and noncommutative geometry. Algebraic geometry is a branch of mathematics studying geometric objects defined by polynomial equations and related mathematical structures. In classical algebraic geometry one associates with such geometric objects (called algebraic varieties) a space of functions that is a commutative ring. In this research project, more sophisticated algebraic structures associated with algebraic varieties, such as A-infinity algebras and derived categories of sheaves, will be studied.The research project will focus on the following topics:1) A-infinity structures associated with curves and their relation to the moduli spaces of curves,2) Semiorthogonal decomposition of the derived categories of equivariant sheaves for finite group actions,3) Cohomological field theories associated with quasihomogeneous polynomials,4) Sheaves on NC-thickenings and a characterization of Jacobians.The first project is about some A-infinity algebras associated with curves with marked points. The research will study normal forms of these A-infinity algebras up to homotopy and to relate them to the moduli spaces of curves. In the second project a construction of a canonical semiorthogonal decomposition of the derived category of equivariant coherent sheaves for some actions of finite reflection groups is outlined and will be studied. The third project is concerned with applications of categories of matrix factorizations with computation in the cohomological field theories attached to quasihomogeneous polynomials with isolated singularities. The fourth project focuses on which coherent sheaves on an abelian variety can be extended to a noncommutative thickening, which is a quantization of the Poisson envelope of the sheaf of regular functions. This may lead to a new characterization of Jacobians of curves.

该研究项目位于代数几何学领域，与弦理论和非交通性几何形状有一些联系。代数几何形状是研究由多项式方程和相关数学结构定义的数学对象的数学分支。在经典的代数几何形状中，一个人与这种几何对象（称为代数品种）相关联，一个函数空间是一个通勤环。 In this research project, more sophisticated algebraic structures associated with algebraic varieties, such as A-infinity algebras and derived categories of sheaves, will be studied.The research project will focus on the following topics:1) A-infinity structures associated with curves and their relation to the moduli spaces of curves,2) Semiorthogonal decomposition of the derived categories of equivariant sheaves for finite小组动作，3）与准多项式相关的共同体学领域理论，4）在NC厚的Hic弹上和Jacobian的表征上的滑轮。第一个项目大约是一些与曲线相关的A-Anfinity代数。该研究将研究这些a-内代数的正常形式，直到同义，并将它们与曲线的模量空间联系起来。在第二个项目中，概述了并将研究对有限反射组的某些作用的衍生类似类别类别的规范半双相分解的结构。第三个项目涉及矩阵因素化类别的应用，并在与孤立的奇异性的准多项式相关的共同体学领域理论中进行了计算。第四个项目的重点是在阿伯利亚品种上连贯的束带可以扩展到非交通增厚，这是对常规功能的托盘的泊松信封的量化。这可能会导致雅各布人的新特征。