Holonomic D-Modules on Abelian Varieties

• Applicant: Professor Dr. Thomas Krämer
• Subject Area: Mathematics
• Term: from 2015 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 269346902

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.

• DFG Programme: Research Fellowships
• International Connection: France
• Host: Professor Dr. Claude Sabbah