On the L2-dbar-cohomology of certain complete Kähler metrics

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 π: M→V be a resolution of V. Under suitable assumptions we show that the (v,q) L2 dbar cohomology of the regular part of V is isomorphic to the (v,q) dbar cohomology of M. Then we show that the previous isomorphism applies to the case of Saper-type Kähler metrics, as introduced by Grant Melles and Milman, and to the case of complete Kähler metrics with finite volume and pinched negative sectional curvatures.