  We introduce filtrations in chiral homology complexes of smooth elliptic
curves, exploiting the mixed Hodge structure on cohomology groups of
configuration spaces. We use these to relate the chiral homology of a smooth
elliptic curve with coefficients in a vertex algebra with the Poisson homology
of the associated Poisson scheme. As an application we deduce finite
dimensionality results for chiral homology in low degrees.
