Converse of \(L^2\) Existence and Extension of Cohomology Classes

The present paper grew out of my talk at Hayama conference in 2022. We give a survey of recent progress on a converse of \(L^2\) existence theorem and a criterion of Nakano positivity which is used to answer a problem of Lempert affirmatively. We also mention a recent result on the extension of cohomology classes which answers a problem of Cao–Demailly–Matsumura affirmatively, and an injectivity theorem as an application.