Singular metrics on vector bundles and a criteria of Nakano spositivity

First we'll recall some recent results on multiplier ideal sheaves and converse $L^2$ theory, including the solution of Demailly's strong openness conjecture on multiplier ideal sheaves (solved by Guan–Zhou) and converses of $L^2$ existence theorems and $L^2$ extensions (established by Deng-Ning–Wang–Zhou), then we'll explain some new results on multiplier submodule sheaves associated to singular hermitian metric on holomorphic vector bundles, including a criteria of holomorphic vector bundles being Nakano semipositivity and a solution of Lempert's problem on Nakano semipositivity.