Frobenius manifolds, free divisors and quiver representations

这个项目关注的是在预齐次向量空间中作为判别式出现的特殊除数。作用于这些向量空间上的约化代数群以规范的方式定义对偶空间上的d模，即所谓的重言系统。如果这个群恰好是一个代数环面，那么这就是众所周知的gkz系统，在这种情况下，开始的除数有法交。这个特定的简单示例解决了该领域的许多有趣问题（至少部分解决了）。该项目的总体目标是研究由预齐次判别式定义的d模的Hodge理论和范畴性质。此外，已建立的gkz系统与环面镜像对称性之间的关系也可以推广到由这种预齐次作用所定义的自系统。我们试图研究这些系统的完整秩，描述这些系统的某些中间扩展（即交上同模），并研究重述系统下混合Hodge模的情况下的Hodge过滤。预齐次判别式的范畴方面将通过描述定义判别式除数的方程的矩阵分解范畴来观察，并将讨论这些范畴的同调镜像对称陈述。所有这些项目的主要困难在于所研究的除数既不是环面除数，也没有孤立的奇点。因此，这些领域的许多传统方法不能直接应用。我们希望在镜像对称陈述中获得超越环的新见解，以及更好地理解预齐次判别式奇点的代数和几何结构。

This project is concerned with particular divisors that appear as discriminants in prehomogeneous vector spaces. The reductive algebraic group acting on those vector spaces defines in a canonical way a D-module on the dual space, a so-called tautological system. If the group happens to be an algebraic torus, then this is nothing but the well-known GKZ-system, in which case the divisor tostart with has normal crossings. Many of the interesting problems in this area are solved (at least partially) for this particular simple example.The overall aim of the project is to study Hodge theoretic as well as categorical properties of D-modules defined by prehomogenous discriminants. Moreover, the well established relation between GKZ-systems and toric mirror symmetry shall be generalized to autological systems defined by such prehomogenous actions. We seek to investigate the holonomic rank of such systems, describe certain intermediate extensions (i.e., intersection cohomology modules) of these systems and study the Hodge filtration in cases where tautological systems underly mixed Hodge modules. The categorical aspects of prehomogeneous discriminants shall be looked at by describing the category of matrix factorization of equations defining a discriminant divisor, and homological mirror symmetry statements for these categories will be discussed. The main difficulty in all these projects is that the divisors under investigation are neither toric, nor do they have isolated singularities. Hence many of the traditional methods in these areas cannot be applied directly. We hope to obtain both new insights in mirror symmetry statements beyond the toric case as well as a better understanding of the algebraic and geometric structure of singularities of prehomogeneous discriminants.