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

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.

本课题属于代数几何领域，与弦理论和非交换几何有一定的联系。代数几何是研究由多项式方程和相关数学结构定义的几何对象的数学分支。在经典代数几何中，人们把这样的几何对象（称为代数变量）与一个函数空间联系起来，即交换环。在本研究项目中，将研究与代数变体相关的更复杂的代数结构，如a -∞代数和派生的束类。主要研究方向为：1)与曲线相关的a -∞结构及其与曲线模空间的关系；2)有限群作用下等变轴的半正交分解；3)与拟齐次多项式相关的上同场理论；4)轴的非线性增厚和雅可比矩阵的表征。第一个课题是关于一些与带标记点的曲线相关的a -∞代数。本研究将研究这些a无穷代数的正规形式直至同伦，并将它们与曲线的模空间联系起来。在第二个项目中，我们给出了有限反射群作用的等变相干束派生范畴的正则半正交分解的构造，并将对其进行研究。第三个项目是关于具有孤立奇点的拟齐次多项式的上同场理论中具有计算的矩阵分解范畴的应用。第四个项目的重点是在阿贝尔变量上的相干束可以扩展到非交换增厚，这是正则函数束的泊松包络的量化。这可能导致曲线雅可比矩阵的一种新的表征。