Holonomic D-Modules on Abelian Varieties

代数几何、分析和拓扑学之间的一个重要联系是通过完整d模对代数微分方程的研究。这导致了Hodge理论和奇点理论的强大扩展，如混合Hodge模块，扭曲结构，tt*几何和与镜像对称的关系。完整的d模可以看作是具有奇点的平连接，而Mochizuki和Sabbah关于扭转模的基本工作为接近任意不规则奇点铺平了道路。在项目中，我们将研究阿贝尔变体的情况，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。出现的群是有趣的动机性质的新不变量，与经典模问题如肖特基问题密切相关。我们的主题是通过扭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，这将有助于更好地理解完整d模的Tannaka群和傅里叶- mukai变换。

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.