On the $L^2$-$\overline{\partial }$-cohomology of certain complete Kähler metrics

Abstract

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 $\pi :M\to V$ be a resolution of $V$. Under suitable assumptions on $h$ we prove that $$H_{2,\overline{\partial }}^{v,q}(\mathrm{reg}(V),g)\cong H_{\overline{\partial }}^{v,q}(M),q=0,...,v.$$ Then we show that the previous isomorphism applies to the case of Saper-type Kähler metrics and to the case of complete Kähler metrics with finite volume and pinched negative sectional curvatures.

arXiv commentFinal version. To appear on Mathematische Zeitschrift