Some L2 results for $\overline\partial$ on projective varieties with general singularities

Let $X$ be an irreducible $n$-dimensional projective variety in ${\Bbb C}P^N$ with arbitrary singular locus. We prove that the $L^2$-$\overline\partial$-$(p,1)$-cohomology groups (with respect to the Fubini-Study metric) of the regular part of $X$ are finite dimensional.