Given a perverse sheaf or a holonomic D-module on an abelian variety there are two ways to associate a set of holomorphic one forms on it one via the singular support and one via the generic vanishing theory. In this talk I will present a joint work with Feng Hao and Yongqiang Liu where we connect these two sets. On a smooth projective irregular variety our results relates to a conjecture proposed by Kotschick and studied by Schreieder and shows that their conjecture can be reinterpreted as follows: the existence of nowhere vanishing holomorphic one forms is equivalent to the non-existence of components given by conormal space of varieties of general type in the decomposition theorem for the albanese morphism. Using some known results we show that the condition is necessary.[-]

The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated categories of regular holonomic D-modules and of constructible sheaves. In a joint work with Masaki Kashiwara, we proved a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and is influenced by Tamarkin's work on symplectic topology. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya.[-]

In their preprint about the Shafarevich conjecture for hypersurfaces on abelian varieties, Lawrence and Sawin prove a big monodromy theorem for families of hypersurfaces by reducing it to a similar result for Tannaka groups of perverse intersection complexes. A large part of their work is an intricate combinatorial argument about Hodge numbers, which is used to exclude that the Tannaka group acts via wedge powers of the standard representation of SL(n). We explain a simple geometric proof of the analogous result when hypersurfaces are replaced by subvarieties of high codimension; this is joint work in progress with Ariyan Javanpeykar, Christian Lehn and Marco Maculan.[-]