Semiclassical Ohsawa–Takegoshi extension theorem and asymptotics of the orthogonal Bergman kernel

We study the asymptotics of Ohsawa–Takegoshi extension operator and orthogonal Bergman projector associated with high tensor powers of a positive line bundle. More precisely, for a fixed complex submanifold in a complex manifold, we consider the operator which associates to a given holomorphic section of a positive line bundle over the submanifold the holomorphic extension of it to the ambient manifold with the minimal $L^2$-norm. When the tensor power of the line bundle tends to infinity, we obtain an explicit asymptotic expansion of this operator. This is done by proving an exponential estimate for the associated Schwartz kernel and showing that this Schwartz kernel admits a full asymptotic expansion. We prove similar results for the projection onto holomorphic sections orthogonal to those which vanish along the submanifold.