Topological objects (such as local systems, perverse sheaves) associated to differential systems are a priori defined over the complex numbers. It is a natural question (motivated, for example, by Hodge theory) to ask under which assumptions these objects can be defined over a smaller field. This talk is about a joint work with Davide Barco, Marco Hien and Christian Sevenheck, where we study this question in the case of hypergeometric D-modules. We consider the problem of Galois descent in the context of the irregular Riemann-Hilbert correspondence of D'Agnolo-Kashiwara, and we show how Betti structures on the solutions of hypergeometrics are obtained by using a geometric realization of these D-modules.
GKZ hypergeometric systems form a well-studied class of D-modules that are related to embedded toric varieties. For certain parameters, they can be constructed functorially from rank one local systems on the underlying torus by a result of Schulze and Walther. This can be used to endow GKZ systems with the structure of a complex mixed Hodge module, as shown by Reichelt. In this talk, I will recall these results and discuss new progress on their extension to tautological systems related to orbit closures of other group representations, in particular in the case of homogeneous spaces. This is ongoing joint work with Thomas Reichelt, Christian Sevenheck, Avi Steiner and Uli Walther.
Generically, the Radon transform transforms a simple nonconstant perverse sheaf on the projective plane concentrated in degree -2 (an intermediate extension of a local system on a dense open subset) to a sheaf on the dual plane, which is again an intermediate extension of a local system placed in degree -2 on a plane curve complement. We show how to explicitly compute the monodromy of this local system.
I will describe some results (joint work with Alberto Castano Dominguez and Luis Narvaez Macarro) on Hodge ideals for a specific class of divisors with non-isolated singularities. Hodge ideals as defined by Mustata and Popa generalize multiplier ideals and are given by the Hodge filtration on the module of meromorphic functions along a divisor. However, they are usually hard to determine. For a certain class of free divisors (e.g. free hyperplane arranngements), one can rely on specific symmetry properties of the Bernstein-Sato polynomial of the divisor, and a basic property of Hodge modules, called strict specializability, to give a purely algebraic description of all Hodge ideals. I will explain this approach, and discuss some significant examples.