  For a projective algebraic variety $V$ with isolated singularities, endowed
with a metric induced from an embedding, we consider the analysis of the
natural partial differential operators on the regular part of $V$. We show
that, in the complex case, the Laplacians of the de Rham and Dolbeault
complexes are discrete operators except possibly in degrees $n,n\pm 1$, where
$n$ is the complex dimension of $V$. We also prove a Hodge theorem on the
operator level and the $L^2$--Stokes theorem outside the degrees $n-1,n$. We
show that the $L^2$-Stokes theorem may fail to hold in the case of real
algebraic varieties, and also discuss the $L^2$-Stokes theorem on more general
non-compact spaces.
