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.
Griffiths and Nakano positivity and Nakano-Nadel vanishing theorems for singular Hermitian metrics on complex spaces
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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 introduce appropriate notions of singular Griffiths/Nakano positivity on complex spaces and establish various properties of these notions. By applying these results, we provide $L^2$-Dolbeault fine resolutions and cohomological isomorphisms, and $L^2$-existence theorems. As an application, we obtain Nakano-Nadel vanishing theorems on weakly pseudoconvex complex spaces.
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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.