Moduli spaces of objects in derived categories

本课题的目的是研究凝聚层导数范畴的一些实例。特别是P.I.有三个主要目的：(1)研究射影平面、K_3曲面和三次四重复数的模空间的几何。(2)把可列几何中的“经典”问题和猜想实现为派生范畴中的穿越现象，第一个例子是射影平面上正则丛的全空间。(3)给出K3曲面的派生范畴与它们的变形之间的等价性的完整描述。本项目更广泛的背景是代数几何领域。作为一门纯数学学科，近年来代数几何与其他科学领域发生了许多相互作用。本项目的动机和灵感正是由菲尔兹奖获得者M.Kontsevich在近20年前提出的，与理论物理和弦理论相结合的。特别是通过从物理中引入直觉和技术来解决纯数学中的经典问题，反之亦然为物理构造提供严谨的数学基础。

The aim of this project is to investigate some instances of the theory of derived categories of coherent sheaves.In particular, the P.I. has three main objectives:(1) To study the geometry of moduli spaces of complexes for the projective plane, K3 surfaces and cubic 4-folds.(2) To realize ``classical'' questions and conjectures in enumerative geometry as wall-crossing phenomena in derived category, the starting example being the total space of the canonical bundle over the projective plane.(3) To give a complete description of equivalences between the derived categories of K3 surfaces and of their deformations.The broader context of this project is the area of algebraic geometry.Starting as a pure mathematical subject, in the recent years algebraic geometry has seen many interactions with other areas of science.The present project is motivated and inspired precisely by connections, envisioned by the Fields Medal winner M. Kontsevich almost twenty years ago, with theoretical physics and string theory.In particular, by bringing intuition and techniques from physics to tackle classical problems in pure mathematics and, vice versa, to provide mathematical rigorous foundations to physics constructions.