Griffiths and Nakano positivity and Nakano-Nadel vanishing theorems for singular Hermitian metrics on complex spaces

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.

v2: 37 pages; added Theorem 1.11, Definition 3.3, Corollary 4.4, and Remark-Theorem 6.2, and made minor revisions. v1: 35 pages. arXiv admin note: text overlap with arXiv:2602.03332