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

布里奇兰稳定物体模空间的穿墙、双有理几何和 Gromov-Witten 理论

• 批准号: 0801356
• 负责人: Arend Bayer
• 金额: $ 9.5万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2010-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801356&HistoricalAwards=false
• 关键词: Wall; Crossing; Moduli; Spaces; Bridgeland

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.

在这个项目中，PI将研究代数簇上相干层的导出类别的稳定性条件。更具体地说，它将考虑在稳定性条件改变（“跨壁”）下稳定对象的模空间的行为。PI将研究跨壁对双有理几何的意义，以及与Donaldson-Thomas和Gromov-Witten不变量相关的枚举问题。模空间的研究在过去的50年里一直是代数几何的驱动力之一。它可以回答关于“对象”-如向量束，曲线，地图-如何在“家庭”中表现的基本问题。 另一方面，在过去的15年里，随着弦理论、辛几何和代数几何的引入，导出范畴的主题有了显著的发展。 本提案的项目将这两项指导原则联合收割机结合起来。 由于穿墙是上述学科普遍感兴趣的现象，研究人员希望为现有的富有成效的跨学科合作做出贡献。