Mixed Hodge structures for vanishing cycles and orbifold cohomology

TL;DR

This paper explores the relationships between vanishing cycles, Milnor rings, and orbifold cohomology rings for Laurent polynomials, establishing isomorphisms and analyzing Hodge structures and dualities in this context.

Contribution

It introduces a mixed Hodge structure on orbifold cohomology rings and examines the polarization and dualities, extending prior work in singularity theory and orbifold cohomology.

Findings

Isomorphisms between vanishing cycles, Milnor rings, and orbifold cohomology.

Identification of mixed Hodge structures on orbifold cohomology.

Analysis of the Hodge-Tate case and duality properties.

Abstract

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.