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

In this paper, in order to develop a more general $L^2$-theory for the $\bar{\mathrm{\partial }}$-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.