Combinatorial Hodge theory in singularity theory and toric geometry

• Applicant: Professor Dr. Christian Sevenheck
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539847082

The principal aim of this project is to advance the understanding of invariants from Hodge theory that one can associate to hyperplane arrangements on the one hand, and to torus actions on the other hand. In both cases, we will be specifically interested in systems of differential equations defined by these Hodge structures. More precisely, for a complex hyperplane arrangement, we shall study the Hodge ideals which determine the Hodge filtration on the module of meromorphic functions along the arrangement. We seek for a combinatorial description of these ideals, and we will study to which extend they are determined by the matroid associated to the arrangement. We shall study the generating level of the Hodge filtration for certain classes of arrangements, and work towards compatibility properties of filtrations on certain cyclic D-modules that govern the Hodge module structure on the sheaf of meromorphic functions along a divisor. We will also develop and implement algorithms to calculate Hodge ideals of arrangements. A second research direction that will be pursued during this project consists in studying combinatorial aspects of the Hodge theory of certain hypergeometric system of differential equations. These include the so-called better behaved GKZ-systems, for which we shall study the duality theory and certain applications to toric mirror symmetry. We will further exploit the relation between arrangements and toric geometry by studying the Hodge structures (e.g. GKZ-systems) defined by the Bergman fan of a matroid, thus linking the two parts of the project. This tight relation should also be exploited when studying the weight filtrations on both meromorphic functions along an arrangement, and on certain GKZ-systems.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2458: Combinatorial Synergies