Equivariant cohomology classes in quiver theory and statistical mechanics, and, a more geometric foundation of intersection theory

颤动理论和统计力学中的等变上同调类，以及交集理论的几何基础

• 批准号: 0956233
• 负责人: Allen Knutson
• 金额: $ 0.02万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-01-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0956233&HistoricalAwards=false
• 关键词: Equivariant; cohomology; classes; quiver; theory

A common feature of my proposals is the use of algebro-geometric degeneration to study an interesting irreducible variety in terms of the many components it breaks into in a limit. Such degenerations are usually very destructive of the geometry, but the original and limiting schemes may define the same (equivariant) cohomology class in some ambient space. The degenerations I intend to study are (1) of moduli spaces of representations of quivers, where the type A case has become well understood in the last three years but types D,E are totally open; (2) of nilpotent orbits in Lie algebras, to the normal cones of their upper triangular part. The homology of the upper triangular part carries a (Springer) representation of the corresponding Weyl group; the normal cone I study has more components and, in the example I can compute so far, carries a representation of a richer algebra; (3) of subvarieties of projective space to reduced schemes with finite maps to projective space, where the degree of the map carries the information more usually seen in the nonreduced structure of the limit subscheme.If we replace a polynomial equation, like xy = zw, by one of its monomials, like xy = 0, the locus of solutions can often be made to retain the same volume even as it breaks into pieces. Where xy=zw has interesting nonflat geometry, but cannot be factored into pieces, xy=0 has interesting discrete structure (it has two pieces, x=0 and y=0, which intersect one another) even though each piece is flat. In the general setup, we have many polynomial conditions to begin with, and each is degenerated" to a simpler polynomial containing only some of the original monomials; a complicated condition guarantees that the dimension and volume of the solution set remains constant. In my work I study these degenerations where the ambient space is a space of (lists of) matrices, and the original equations are natural matrix conditions like "matrices whose square is zero". The difficulty comes in finding degenerations that preserve the dimension and volume, and then studying the resulting equations to understand the simpler pieces into which the degenerate locus factors. The resulting discrete structure is then of great interest combinatorially, and sheds light on the (topologically important) volume of the original locus.

我的建议的一个共同特点是使用代数几何退化来研究一个有趣的不可约品种的许多组成部分，它打破了限制。这样的退化通常对几何有很大的破坏性，但是原始的和极限的方案可以在某些周围空间中定义相同的（等变的）上同调类。我打算研究的退化是：（1）箭图表示的模空间，其中A型情形在过去三年中已得到很好的理解，但D、E型情形是完全开放的;（2）李代数中幂零轨道退化到其上三角部分的正规锥。上三角形部分的同源性带有一个（施普林格）表示相应的Weyl群;正常锥我研究有更多的组件，并在例子中，我可以计算到目前为止，携带一个更丰富的代数表示;（3）射影空间的子簇到射影空间的有限映射的约化概型，其中映射的次数携带了在极限子方案的非约化结构中更常见的信息。如果我们将多项式方程（如xy = zw）替换为它的一个单项式（如xy = 0），通常可以使溶液的轨迹保持相同的体积，即使它破碎成碎片。 其中xy=zw有有趣的非平坦几何，但不能分解成碎片，xy=0有有趣的离散结构（它有两个碎片，x=0和y=0，它们彼此相交），即使每个碎片都是平坦的。 在一般的设置中，我们有许多多项式条件开始，并且每个条件都退化为一个只包含一些原始单项式的简单多项式;一个复杂的条件保证解集的维数和体积保持不变。 在我的工作中，我研究这些退化的环境空间是一个（列表）矩阵的空间，和原始方程是自然的矩阵条件，如“矩阵的平方为零”。困难在于找到保持尺寸和体积的简并，然后研究所得方程以理解简并轨迹因子的更简单的部分。由此产生的离散结构，然后组合的极大兴趣，并揭示了（拓扑重要的）体积的原始轨迹。