Derived categories techniques in algebraic geometry

代数几何中的派生范畴技术

• 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

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.

本论文的研究将集中在三个主题上，都是应用代数几何中的导出范畴的技巧：1）与孤立超曲面奇点相关的上同调场论;2）与交换簇相关的导出范畴的真实的乘法函子和稳定空间; 3）Lagrange Grassmannian算子的例外集合.第一个项目是构造一个与孤立的拟Lagrange Grassmannian算子相关的上同调场论，本质上，这相当于在具有在模空间的边界上满足某些因子分解规则的标记点的稳定曲线的模空间上构造上同调类的集合，第二个项目是研究阿贝尔簇的导出范畴之间的某些函子，这些函子可以用于Manin的真实的乘法程序中。第三个方案是在辛向量空间中的Lagrange子空间的Grassmannian上构造出凝聚层的导出范畴中的向量丛的完全例外集，这种例外集由一定的上同调条件定义，通过将几何问题转化为线性代数问题，促进了代数簇上相干层的研究。拟议的研究是在代数几何领域，与弦理论和非交换理论有一定的联系。几何代数几何是数学的一个经典分支，研究由多项式方程和相关数学概念定义的几何对象。代数几何中涉及模空间（对各种几何结构进行分类的参数空间）的某些部分的许多最新进展都是由它们在弦理论中的应用所激发的。导出范畴产生于对链复形范畴的研究，并形成了现代代数几何所需的庞大代数机器的一部分。