Big line bundles on compact complex spaces satisfy a Steenbrink-type cohomology vanishing theorem after passing to a log resolution and twisting by multiplier ideal sheaves.
$L^2$-Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics
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abstract
In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we provide $L^2$-Dolbeault fine resolutions and isomorphisms, and $L^2$-estimates, for holomorphic line bundles on complex spaces equipped with singular Hermitian metrics. As applications, we obtain several generalizations of the Nadel vanishing theorem.
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Steenbrink vanishing theorem for big line bundles
Big line bundles on compact complex spaces satisfy a Steenbrink-type cohomology vanishing theorem after passing to a log resolution and twisting by multiplier ideal sheaves.