Moduli spaces of morphisms and sheaves

态射和滑轮的模空间

• 批准号: 0852468
• 负责人: Dragos Oprea
• 金额: $ 7.36万
• 依托单位: University of California-San Diego
• 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 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0852468&HistoricalAwards=false
• 关键词: Moduli; spaces; morphisms; sheaves

The PI of this project will study questions related to the moduli spaces of sheaves and morphisms. Regarding the moduli spaces of sheaves, the main themes to be covered are: the 'strange duality' isomorphism for the moduli space of bundles on a curve as the curve varies in moduli - in particular, extensions of 'strange duality' to nodal curves and to general structure groups; the Theta bundles, their sections, and 'rank-level' dualities on the moduli spaces of sheaves on surfaces (the latter in collaboration with Alina Marian); computation of intersection theoretic invariants on the moduli space of higher-rank framed/unframed sheaves on threefolds. Regarding stable morphisms, and building on previous work, the author will pursue a further study of the topology of the moduli space of stable maps and of the associated tautological systems. The projects above are aimed to the understanding of the topology, geometry and intersection theory of various moduli spaces of sheaves and morphisms. One of the goals is to exploit the inter-connections these moduli spaces enjoy.Moduli spaces arise as answers to classification and parametrization problems, putting together objects with similar structure. Questions about these objects (e.g. counting problems, questions about families) often involve a detailed study of the relevant moduli spaces. The moduli spaces of stable sheaves and stable morphisms are two of the primary examples occurring in algebraic geometry. In addition to their importance within the field, they bear multiple connections with other branches of classical and modern mathematics such as complex analysis, representation theory, theoretical physics, symplectic geometry, gauge theory, differential geometry, combinatorics and topology.

这个项目的PI将研究与束模空间和态射相关的问题。关于束的模空间，要讨论的主题是：曲线上束模空间随模量变化的“奇异对偶性”同构——特别是“奇异对偶性”在节点曲线和一般结构群上的推广；Theta束，它们的截面，以及曲面上束的模空间上的“秩-水平”对偶性（后者是与Alina Marian合作的）；三层高阶有框/无框轮轴模空间上交点理论不变量的计算。关于稳定态射，在前人工作的基础上，作者将进一步研究稳定映射的模空间和相关的同义系统的拓扑。上述项目旨在了解轮轴和模态的各种模空间的拓扑、几何和交点理论。其中一个目标是利用这些模空间所享有的相互联系。模空间作为分类和参数化问题的答案而出现，将具有相似结构的对象放在一起。关于这些对象的问题（例如计数问题，关于家庭的问题）通常涉及对相关模空间的详细研究。稳定束和稳定态射的模空间是代数几何中出现的两个主要例子。除了它们在该领域的重要性之外，它们还与古典和现代数学的其他分支，如复分析、表示理论、理论物理、辛几何、规范理论、微分几何、组合学和拓扑学等，有着多种联系。