Hodge Structures for Orbifold Cohomology

We construct a polarized Hodge structure on the primitive part of Chen and Ruan’s orbifold cohomology Hk orb(X) for projective SL-orbifolds X satisfying a “Hard Lefschetz Condition”. Furthermore, the total cohomology ⊕ p,q H p,q orb(X) forms a mixed Hodge structure that is polarized by every element of the Kähler cone of 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. 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, in light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.