  Let $(M,g)$ be an incomplete Riemannian manifold of finite volume and let
$2\leq p<\infty$. In the first part of this paper we prove that under certain
assumptions the inclusion of the space of $L^p$-differential forms into that of
$L^2$-differential forms gives rise to an injective/surjective map between the
corresponding $L^p$ and $L^2$ cohomology groups. Then in the second part we
provide various applications of these results to the curvature and the
intersection cohomology of compact Thom-Mather stratified pseudomanifolds and
complex projective varieties with only isolated singularities.
