  Let $V$ be a complex projective variety with isolated singularities. Let the
smooth part be given the metric induced by a projective imbedding. Then we
develop the $L_2$ harmonic theory and construct a pure Hodge structure on the
$L_2$-cohomology of $V$. If the dimension of $V$ is two, we put a cohomological
Hodge structure on its complex of sheaves of $L_2$ forms.
