  Let $V$ be a compact and irreducible complex space of complex dimension $v$
whose regular part is endowed with a complete Hermitian metric $h$. Let
$\pi:M\rightarrow V$ be a resolution of $V$. Under suitable assumptions on $h$
we prove that $$H^{v,q}_{2,\overline{\partial}}(\operatorname{reg}(V),g)\cong
H^{v,q}_{\overline{\partial}}(M),\ q=0,...,v.$$ Then we show that the previous
isomorphism applies to the case of Saper-type K\"ahler metrics and to the case
of complete K\"ahler metrics with finite volume and pinched negative sectional
curvatures.
