Mixed Hodge structures for vanishing cycles and orbifold cohomology

Above a Laurent polynomial f one makes grow a vector space of vanishing cycles (after the work of Sabbah, singularity setting), a graded Milnor ring (after the work of Kouchnirenko) and an orbifold cohomology ring (after the work of Borisov, Chen and Smith). Under suitable assumptions, these structures are isomorphic and these identifications are interesting because some results are more explicit in one setting than in another. In particular, and in order to understand better the real structures and the dualities appearing in the singularity setting, we first look for the counterpart of Sabbah's mixed Hodge structures, initially defined on the space of vanishing cycles, on the orbifold cohomology ring. Then, we discuss to what extent the orbifold Poincar\'e duality defined by Chen and Ruan provides a polarization of this mixed Hodge structure. We study in details the Hodge-Tate case, which can be read off from the ages of the sectors, a variation of the hard Lefschetz condition introduced by Fernandez. These notes go along with prior works of Fernandez and Wang.