Résume | Ohsawa and Takegoshi in 1987 gave a sufficient condition under which a holomorphic section of a vector bundle on the submanifold extends to the holomorphic section over the ambient manifold. In this talk, in the semiclassical setting, i.e. when the section is taken from a suffiently big tensor power of a positive line bundle, we provide an explicit asymptotic formula for the optimal extension operator. We derive several consequences, among which are the asymptotic transitivity of the extension operator with respect to the tower of submanifolds, calculation of the asymptotic of the optimal constant in Ohsawa-Takegoshi extension theorem and higher derivative bounds on holomorphic extensions, recently asked by Demailly in his conjectural approach to the invariance of plurigenera for Kähler families. |