Geometry and Topology of Moduli Spaces of Parabolic Bundles, Toric Varieties, and Partial Flag Manifolds

抛物线丛、环面簇和部分旗流形的模空间的几何和拓扑

• 批准号: 0072520
• 负责人: Philip Foth
• 金额: $ 7.47万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072520&HistoricalAwards=false
• 关键词: Geometry; Topology; Moduli; Spaces; Parabolic

DMS-0072520 Philip Foth The PI will study the topology of certain algebraic varieties by methods of Poisson geometry. Many interesting varieties admit Poisson structures such that the closures of the symplectic leaves for such structures additively generate the homology. Moreover, one can ask for certain quite strong invariance and asymptotic properties of thosePoisson structures so that the complete information about the (equivariant) cohomology of these varieties may be obtained by studying the corresponding Poisson harmonic forms. In addition, if the algebraic variety in question is smooth and has an invariant Kaehler structure such that the Kaehler form is compatible in a certain sense with the Poisson structure, then one can hope to find such Poisson structures on the GIT quotients of the variety using the correspondence between the symplectic and GIT quotients. In addition, one can hope to study the Chow quotients by similar methods. One also hopes to obtain new integrable systems in this context or throw a new light on already known ones. The varieties which fall into the scope of interest of this project include (partial) flag manifolds, the moduli spaces of parabolic bundles, toric varieties, and others. There is also a connection between these topics and formal geometry and deformation quantization. The PI would like to understand the topology (underlying structure) of certain spaces that appear in different branches of science such as geometry and theoretical physics. The results that PI intends to obtain are likely to have applications to topological, conformal, and quantum field theories, as well as strings and mirror symmetry. In these endeavors the PI plans to apply methods of algebraic and differential geometry and the formal algebraic apparatus. The spaces under investigation come naturally equipped with rich algebraic and geometric structures. Some of these spaces appear as quite classical objects (like grassmannians) and some of them are more sophisticated (like moduli spaces). The PI is looking forward to further unveiling these structures and relating them to known phenomena sometimes crossing interdisciplinary borders.

DMS-0072520 Philip Foth PI将通过泊松几何的方法研究某些代数簇的拓扑。 许多有趣的品种承认泊松结构，这样的结构的辛叶的闭包相加产生的同源性。 此外，人们可以要求某些相当强的不变性和渐近性质thosePoisson结构，使完整的信息（等变）这些品种的上同调可以通过研究相应的泊松调和形式。 此外，如果所讨论的代数簇是光滑的，并且具有不变的Kaehler结构，使得Kaehler形式在某种意义上与Poisson结构相容，那么人们可以希望使用辛和GIT代数之间的对应关系在簇的GIT代数上找到这样的Poisson结构。 此外，人们可以希望通过类似的方法来研究Chow事件。 人们还希望在这种情况下获得新的可积系统，或对已知系统提供新的见解。 属于本项目感兴趣的范围的品种包括（部分）旗流形，抛物丛的模空间，环面品种，和其他。 这些主题与形式几何和变形量子化之间也有联系。PI希望了解出现在不同科学分支（如几何学和理论物理学）中的某些空间的拓扑结构（底层结构）。PI打算获得的结果很可能应用于拓扑、共形和量子场论，以及弦和镜像对称。在这些努力中，PI计划应用代数和微分几何的方法以及正式的代数仪器。被调查的空间自然配备了丰富的代数和几何结构。 这些空间中的一些看起来像非常经典的对象（如格拉斯曼空间），而其中一些则更复杂（如模空间）。PI期待着进一步揭示这些结构，并将它们与有时跨越跨学科边界的已知现象联系起来。