Frobenius manifolds, free divisors and quiver representations

本课题主要研究准齐次向量空间中作为判别式出现的特殊因子。作用在这些向量空间上的约化代数群以规范的方式定义了对偶空间上的D-模，即所谓的重言式系统。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，在这种情况下，除数一开始就有正常的交叉点。这个简单的例子解决了这个领域中许多有趣的问题(至少部分地)。本项目的总体目标是研究由预齐次判别式定义的D-模的Hodge理论以及范畴性质。此外，GKZ-系统与环面镜面对称性之间的良好关系将被推广到由这种预齐次作用定义的自体系统。我们研究了这类系统的完整秩数，刻画了这类系统的某些中间扩张(即交上同调模)，并研究了重言式系统欠混合Hodge模的情形下的Hodge滤子。预齐次判别式的范畴方面应通过描述定义判别因子的方程的矩阵因式分解范畴来查看，并且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，被研究的因子既不是环面的，也不是孤立的奇点。因此，这些领域中的许多传统方法不能直接应用。我们希望在超越环面情形的镜像对称陈述中获得新的见解，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.