Ross-Witt Nyström correspondence and Ohsawa-Takegoshi extension

We obtain a general Ohsawa-Takegoshi extension theorem by using the Ross-Witt Nyström correspondence picture and Berndtsson's theorem in \cite{Bern20}. In the test configuration ($\mathbb C^*$-degeneration) case, our approach gives a quick proof of the Ohsawa-Takegoshi extension theorem without taking limit or using singular weight, which is very different from the Ohsawa-Chen-Blocki-Guan-Zhou approach and the Berndtsson-Lempert approach. Another advantage of our approach is that it fits better to the sharp estimate for the weighted Bergman kernel on compact manifold. Applications include a sharp lower bound of the Bergman kernel for compact Riemann surfaces and a non-vanishing theorem in terms of (weighted) Okounkov bodies.