In the present paper, we study the properties of singular Nakano positivity of singular Hermitian metrics on holomorphic vector bundles, and establish an optimal L² extension theorem for holomorphic vector bundles with singular Hermitian metrics on weakly pseudoconvex Kähler manifolds, which is a unified version of the optimal L² extension theorems for holomorphic line bundles with singular Hermitian metrics of Guan–Zhou and Zhou–Zhu. As applications, we give a necessary condition for the holding of the equality in optimal L² extension theorem, and present singular Hermitian holomorphic vector bundle versions of some L² extension theorems with optimal estimate.

In this article, we consider Bergman kernels with respect to modules at boundary points, and obtain a log-subharmonicity property of the Bergman kernels, which implies a concavity property related to the Bergman kernels. As applications, we reprove the sharp effectiveness result related to a conjecture posed by Jonsson–Mustaţă and the effectiveness result of strong openness property of the modules at boundary points.

In this article, we present characterizations of the concavity property of minimal L² integrals degenerating to linearity in the case of products of analytic subsets on products of open Riemann surfaces. As applications, we obtain characterizations of the holding of equality in optimal jets L² extension problem from products of analytic subsets to products of open Riemann surfaces, which implies characterizations of the product versions of the equality parts of Suita conjecture and extended Suita conjecture, and the equality holding of a conjecture of Ohsawa for products of open Riemann surfaces.

In this article, we present the concavity of the minimal $L^2$ integrals related to multiplier ideals sheaves on Stein manifolds. As applications, we obtain a necessary condition for the concavity degenerating to linearity, a characterization for 1-dimensional case, and a characterization for the equality in 1-dimensional optimal $L^2$ extension problem to hold.