  We construct a polarized Hodge structure on the primitive part of Chen and
Ruan's orbifold cohomology $H_{orb}^k(X)$ for projective $SL$-orbifolds $X$
satisfying a ``Hard Lefschetz Condition''.
  Furthermore, the total cohomology $H_{orb}^{*,*}(X)$ forms a mixed Hodge
structure that is polarized by every element of the K\"ahler cone of $X$. Using
results of Cattani-Kaplan-Schmid this implies the existence of an abstract
polarized variation of Hodge structure on the complexified K\"ahler 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 the light of the conjectural
correspondence between the (quantum) orbifold cohomology and the (quantum)
cohomology of a crepant resolution.
