  Let $G$ be an abelian group acting on a smooth algebraic variety $X$. We
investigate the product structure and the bigrading on the cohomology of
polyvector fields on the orbifold $[X/G]$, as introduced by C\u{a}ld\u{a}raru
and Huang. In this paper we provide many new examples given by quotients of
Fermat hypersurfaces, where the product is shown to be associative. This is
expected due to the conjectural isomorphism at the level of algebras between
the cohomology of polyvector fields and Hochschild cohomology of orbifolds. We
prove this conjecture for Calabi-Yau Fermat hypersurface orbifold. We also show
that for Calabi-Yau orbifolds, the multiplicative bigrading on the cohomology
of polyvector fields agrees with what is expected in homological mirror
symmetry.
