Wall-Crossing for Moduli Spaces of Bridgeland-Stable Objects, Birational Geometry and Gromov-Witten Theory

在这个项目中，PI将研究代数品种上相干束的派生范畴的稳定性条件。更具体地说，它将考虑稳定对象的模空间在稳定条件变化（“过壁”）下的行为。PI将研究跨墙对两国几何的意义，以及与Donaldson-Thomas和Gromov-Witten不变量相关的列举问题。在过去的50年里，模空间的研究一直是代数几何的推动力之一。它可以回答关于“对象”（如矢量束、曲线、地图）在“族”中的行为的基本问题。另一方面，随着弦理论、辛几何和代数几何的投入，派生范畴的主题在过去15年中有了显著的发展。本提案的项目结合了这两个指导原则。由于跨界是上述各学科普遍感兴趣的现象，研究者希望对现有的卓有成效的跨学科合作做出贡献。

In this project, the PI will study stability conditions on the derived categories of coherent sheaves on algebraic varieties. More specifically, it will consider the behavior of moduli spaces of stable objects under a change of the stability conditions (``wall-crossing''). The PI will study the significance of wall-crossing to birational geometry, and to enumerative questions related to Donaldson-Thomas and Gromov-Witten invariants. The study of moduli spaces has been one of the driving forces of algebraic geometry in the last 50 years. It can give answers to fundamental questions about how ``objects''---such as vector bundles, curves, maps---behave in ``families''. On the other hand, the topic of derived categories has seen a remarkable development in the last 15 years with input from string theory, symplectic geometry and algebraic geometry. The projects of this proposal combine these two guiding principles. Since wall-crossing is a phenomenon of general interest to each of the above-mentioned disciplines, the investigator hopes to contribute to the existing fruitful interdisciplinary collaboration.