Derived categories techniques in algebraic geometry

本研究将集中在三个主题上，所有这些主题都应用了代数几何中的派生范畴技术：1)与孤立超曲面奇点相关的上同场理论；2)与阿贝尔变相关的派生范畴的实乘法函子和稳定性空间；3)关于拉格朗日格拉斯曼量的特殊收藏。第一个课题是建立一个与孤立拟齐次超曲面奇点相关的上同场理论。本质上，这相当于在稳定曲线的模空间上构造一组上同调类，这些稳定曲线的标记点在模空间的边界上满足某些分解规则，如Gromov-Witten不变量理论。第二个课题是研究可用于非交换环面上的Manin实乘法规划的阿贝尔变体派生范畴之间的某些函子。本文还提出了定义和研究这些函子在bridgeeland稳定空间上的作用。第三个课题是在辛向量空间的格拉斯曼拉格朗日子空间上构造相干束的派生范畴中的向量束的完全例外集合。这种集合由某些上同调条件定义，通过将几何问题转化为线性代数问题，促进了代数变量上相干束的研究。本文的研究方向是代数几何，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概念定义的几何对象的数学的一个经典分支。在代数几何的某些领域中，模空间（对各种几何结构进行分类的参数空间）最近取得的许多进展都是由它们在弦理论中的应用所推动的。衍生范畴源于对链式配合物范畴的研究，构成了现代代数几何所需的庞大代数机制的一部分。

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.