Moduli spaces of objects in derived categories

本项目的目的是研究相干轴的派生范畴理论的一些实例。特别是，P.I.有三个主要目标：(1)研究射影平面，K3曲面和立方4折叠的复合体的模空间的几何形状。(2)将数列几何中的“经典”问题和猜想作为推导范畴中的跨壁现象来实现，以射影平面上正则束的总空间为起始例子。(3)给出了K3曲面及其变形的派生范畴之间的等价性的完整描述。这个项目更广泛的背景是代数几何领域。代数几何最初是一门纯数学学科，近年来与其他科学领域有了许多互动。目前这个项目的动机和灵感正是来自于20年前菲尔兹奖得主M. Kontsevich与理论物理学和弦理论之间的联系。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，为物理结构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.