Abstract : 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.

Keywords : holomorphic one-form; cohomology jump loci; constructible sheaves; singular support