Complex geometry of noncommutative tori and t-structures on derived categories

派生范畴上非交换环面和 t 结构的复杂几何

基本信息

批准号：
0601034
负责人：
Alexander Polishchuk
金额：
$ 12.7万
依托单位：
University of Oregon Eugene
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2006
资助国家：
美国
起止时间：
2006-06-01 至 2010-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0601034&HistoricalAwards=false
关键词：
Complex geometry noncommutative tori structures

项目摘要

The proposed research will focus mainly on two topics:1) holomorphic bundles on noncommutative tori, and 2) t-structures and stability conditions on derived categories.Noncommutative tori play an important role in noncommutative geometry being the simplest examples of noncommutative manifolds. However, putting the complex structure and considering corresponding holomorphic objects on them is a recently new idea. The first project is devoted to the study of the categories of holomorphic bundles on noncommutative tori generalizingknown results in the two-dimensional case. The second project is concerned withinteresting new structures on derived categories of coherent sheaves on algebraic varietiesdiscovered by Bridgeland (motivated by some ideas from physics). Namely, the goalis to attack a number of problems related to stability conditions on such derived categories.Two more projects in the proposal are concerned with the study of tautological cycles on Jacobian varieties and with A-infinity algebras arising in algebraic geometry.The proposed research is in the fields of noncommutative geometry and algebraic geometry.Noncommutative geometry is a relatively new field originally developed by Connes that combines ideas from noncommutative algebra, differential geometry and functional analysis. Algebraic geometry is a classical branch of mathematics studying geometric objects defined by polynomial equations and related mathematical concepts. Many recent advances in both fieldswere motivated by their use in physics.

拟议的研究将主要集中在两个主题上：1）对非交易性托里的全态束，以及2）T结构和稳定条件对派生类别的稳定性。非共同的Tori在非交互性几何学中起重要作用，是非交换歧管的最简单实例。但是，将复杂的结构放在它们上，考虑到相应的全态物体是一个新想法。第一个项目致力于研究二维情况下非交易性托里概括已知结果的全体形态束类别的研究。第二个项目涉及在Bridgeland发现的代数品种上的相干滑轮类别的新结构（由物理学的某些想法激励）。就是说，攻击此类派生类别上与稳定性条件有关的许多问题的攻击目的是，该提案中有更多项目与对雅各比式品种的重言式循环的研究以及代数几何形状引起的一个内代代数有关。在拟议的几何学中，提出的研究是在非标准的几何学和Alge egeementer ancortratival Isement ancommential Geementry a的领域中。最初由Connes开发的领域，结合了来自非共同代数，差异几何形状和功能分析的思想。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。两种现场的最新进展都取决于它们在物理学中的使用。