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

本论文的研究主要集中在两个方面：1）非对易环面上的全纯丛; 2）导出范畴上的t-结构和稳定性条件。非对易环面作为非对易流形的最简单例子，在非对易几何中起着重要作用。然而，把复杂结构和考虑相应的全纯对象放在它们上面是最近才出现的新思想。第一个项目是研究非交换环面上的全纯丛范畴，推广了二维情形下的已知结果。第二个项目是关于Bridgeland发现的代数变量上的相干层的导出范畴的有趣的新结构（受物理学的一些思想的启发）。也就是说，目标是解决与这类导出范畴上的稳定性条件有关的一些问题。建议中的另外两个项目涉及Jacobian簇上的重言式循环的研究和A-无穷代数出现在代数几何。拟议的研究是在非交换几何和代数几何领域。非交换几何是一个相对较新的领域，最初由康纳斯，结合思想非交换代数微分几何和泛函分析 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。这两个领域的许多最新进展都是由它们在物理学中的应用所激发的。

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.