Recent progress in the theory of functions of several complex variables and complex geometry

We give a survey on recent progress on converses of $L^2$ existence theorem and $L^2$ extension theorem which are two main parts in $L^2$-theory, and their applications in getting criteria of Griffiths positivity and characterizations of Nakano positivity of (singular) Hermitian metrics of holomorphic vector bundles, as well as the strong openness property and stability property of multiplier submodule sheaves associated to singular Nakano semipositive Hermitian metrics on holomorphic vector bundles.