Hodge structures for orbifold cohomology

Abstract

We construct a polarized Hodge structure on the primitive part of Chen and Ruan’s orbifold cohomology H o r b k ( X ) H_{orb}^k(X) for projective S L SL -orbifolds X X satisfying a “Hard Lefschetz Condition”. Furthermore, the total cohomology ⨁ p , q H o r b p , q ( X ) \bigoplus _{p,q}H_{orb}^{p,q}(X) forms a mixed Hodge structure that is polarized by every element of the Kähler cone of X X . Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified Kähler cone of X X . This construction should be considered as the analogue of the abstract polarized variation of Hodge structure that can be attached to the singular cohomology of a crepant resolution of X X , in light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.