Derived categories techniques in algebraic geometry

代数几何中的派生范畴技术

基本信息

批准号：
1001364
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：
2010
资助国家：
美国
起止时间：
2010-07-15 至 2014-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001364&HistoricalAwards=false
关键词：
Derived categories techniques algebraic geometry

项目摘要

The proposed research will focus on three topics, all applying the techniques of derived categories in algebraic geometry:1) cohomological field theories associated with isolated hypersurface singularities;2) real multiplication functors and stability spaces for derived categories related to abelian varieties;3) exceptional collections on Lagrangian Grassmannians.The first project is to construct a cohomological field theory associated with an isolated quasi-homogeneous hypersurface singularity.Essentially, this amounts to constructing a collection of cohomology classes on the moduli spaces of stable curves with marked points that satisfy certain factorization rules over the boundary of the moduli spaces, as in the theory of Gromov-Witten invariants.The second project is to study certain functors between derived categories of abelian varieties that can be used in Manin's real multiplication program for noncommutative tori. It is also proposed to define and study the action of these functors on the Bridgeland's stability spaces.The third project is to construct full exceptional collections of vector bundles in the derived category of coherent sheaves on the Grassmannian of Lagrangian subspaces in a symplectic vector space.Such collections, defined by certain cohomological conditions, facilitate the study of coherent sheaves on an algebraic variety by transferring geometric questions into linear algebra problems.The proposed research is in the field of algebraic geometry with some connections to string theory and noncommutative geometry. Algebraic geometry is a classical branch of mathematics studying geometric objects defined by polynomial equations and related mathematical concepts. Many recent advances in some parts of algebraic geometry involving moduli spaces (parameter spaces classifying various geometric structures) were motivated by their use in string theory. Derived categories arose from studying categories of chain complexes and form a part of a vast algebra machinery needed for modern algebraic geometry.

拟议的研究将重点介绍三个主题，所有主题都应用了代数几何形状中的衍生类别技术：1）与孤立的高度表面奇异性相关的共同体田地理论； 2）与阿比亚族相关的界限相关的典型派系的划分的派出类别；本质上，这相当于在稳定曲线的模量空间上构建一系列的共同体学课，并带有明显的点，可以满足对模量空间边界的某些分解规则的规则，就像在gromov-witten无害的理论中，在gromov-witten无害的理论中都可以在某些范围内进行派生的范围。非交换托里。还建议它定义和研究这些函数对bridgeland的稳定空间的作用。第三个项目是在派生的沿Lagrangian子空间的衍生式吊带板上的派生类别中构建全面的矢量捆绑包，在符号矢量空间中，在一定的组合中定义了一个综合的研究，该群体在整个群体中定义了整个研究。将几何问题转移到线性代数问题中。拟议的研究在代数几何学领域，与字符串理论和非交通性几何形状有所连接。代数几何形状是数学的经典分支，研究了由多项式方程和相关数学概念定义的几何对象。在涉及模量空间（参数空间进行各种几何结构的参数空间）的代数几何形状的某些部分中，许多最新进展是由它们在字符串理论中使用的动机。派生类别来自研究链复合物的类别，并构成现代代数几何形状所需的巨大代数机械的一部分。