Intersection Theory and Geometric Dualities on Moduli Spaces of Sheaves

DMS-0401670 Serguei Arkhipov该项目将在三个层次上研究半无限上同调和半正则模如何集成到各种代数几何结构中：对于有限维对象(包括标志簇、幂零锥和半单代数群的Springer簇)，对于由于Beilinson和Drinfeld而在几何朗兰兹计划过程中出现的对象(仿射Grassmannians和仿射标志簇)，以及最后，对于与二维局部场相连的对象，例如具有有限维李代数中的值的迭代形式劳伦级数的李代数。因此，我们提出了一个问题层次，其统一的思想是半无限上同调和半正则模。在第一部分中，我们给出了一个关于(非分次)结合代数的半正则双模和半无限上同调的新方法。第二部分处理有限维对象。对半单李代数上的修正半正则模进行了几何刻画。第三部分将小量子群的半无限上同调与仿射旗簇上的倒序、朗兰兹对偶群的Springer簇上的等变凝聚层以及仿射李代数在临界水平上的经典半无限上同调联系起来。第四部分是对取值于半单李代数的迭代Laurent级数的代数构造范畴半正则模的尝试。

DMS-0401670Serguei ArkhipovThe project will investigate how semi-infinite cohomology and semi-regular modules integrate into various algebro-geometric constructions on three levels: for finite dimensional objects (including flag varieties, nilpotent cones and Springer varieties of semi-simple algebraic groups), for objects arising in the geometry of loop groups (affine Grassmannians and affine flag varieties) in the course of Geometric Langlands Program due to Beilinson and Drinfeld and, finally, for objects connected with 2-dimensional local fields, e.g. Lie algebras of iterated formal Laurent series with values in finite dimensional Lie algebras. Thus we propose a hierarchy of problems whose unifying ideas are semi-infinite cohomology and semi-regular modules.The proposed project is divided naturally in four parts. In the first part we work out a certain new approach to semi-regular bimodules and semi-infinite cohomology of (nongraded) associative algebras. The second part deals with finite dimensional objects. We describe geometrically the modified semi-regular module over a semi-simple Lie algebra. Then we provide a geometric description for semi-infinite cohomology of the tensor square of the small quantum group at a root of 1. The third part connects semi-infinite cohomology of small quantum groups with perverse sheaves on the affine flag variety, equivariant coherent sheaves on the Springer variety for the Langlands dual group and, finally, with the classical semi-infinite cohomology of the affine Lie algebra at the critical level. The fourth part is an attempt to construct a categorical semi-regular module for the algebra of iterated Laurent series with values in a semi-simple Lie algebra.