Holonomic D-Modules on Abelian Varieties

代数几何、分析和拓扑学之间的一个重要纽带是通过完整D-模研究代数微分方程。这导致了Hodge理论和奇点理论的强大扩展，如混合Hodge模、扭曲结构、TT*-几何以及与镜像对称性的关系。完整D-模可以看作是与奇点的平坦联系，Mochizuki和Sabbah在旋转模上的基本工作为逼近任意不规则奇点铺平了道路。在这个项目中，我们将研究交换簇的情况，其中群结构导致完整D-模和某些代数群的表示之间的Tannakian对应。产生的群是有趣的新的动机性质的不变量，与经典的模问题密切相关，如肖特基问题。我们的主旨是通过扭曲模理论将这些群与傅里叶-Mukai变换的微分几何性质联系起来，这将导致对完整D-模的Tannaka群和傅里叶-Mukai变换的更好的理解。

An important link between algebraic geometry, analysis and topology is the study of algebraic differential equations via holonomic D-Modules. This leads to powerful extensions of Hodge theory and singularity theory such as mixed Hodge modules, twistor structures, tt*-geometry and relations with mirror symmetry. Holonomic D-modules may be viewed as flat connections with singularities, and the fundamental work of Mochizuki and Sabbah on twistor modules paves the way for approaching arbitrary irregular singularities. In the project we will study the case of abelian varieties, where the group structure leads to a Tannakian correspondence between holonomic D-modules and representations of certain algebraic groups. The arising groups are interesting new invariants of a motivic nature, closely related to classical moduli questions such as the Schottky problem. Our leitmotif will be to connect these groups with differential-geometric properties of the Fourier-Mukai transform via the theory of twistor modules, which will result in a better understanding of both the Tannaka groups and the Fourier-Mukai transform for holonomic D-modules.