Moduli spaces of objects in derived categories

这个项目的目的是研究相干层的导出范畴理论的一些实例。主要有三个目的：（1）研究射影平面、K_3曲面和三次4-折叠复形的模空间几何。(2)将计数几何中的“经典”问题和命题实现为导出范畴中的穿墙现象，起始例子是射影平面上的正则丛的全空间。(3)给出K3曲面及其变形的导出范畴之间的等价性的完整描述。这个项目的更广泛的背景是代数几何领域。作为一个纯数学学科开始，近年来代数几何已经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来自菲尔兹奖赢家M. Kontsevich在近20年前，与理论物理学和弦理论。特别是，通过从物理学中引入直觉和技术来解决纯数学中的经典问题，反之亦然，为物理学的构建提供严格的数学基础。

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.