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

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