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In a setting of a complex manifold with a fixed positive line bundle and asubmanifold, we consider the optimal Ohsawa-Takegoshi extension operator,sending a holomorphic section of the line bundle on the submanifold to theholomorphic extension of it on the ambient manifold with the minimal$L^2$-norm. We show that for a tower of submanifolds in the semiclassicalsetting, i.e. when we consider a large tensor power of the line bundle, theextension operators satisfy transitivity property modulo some small defect,which can be expressed through Toeplitz type operators. We calculate the firstsignificant term in the asymptotic expansion of this "transitivity defect". Asa byproduct, we deduce the composition rules for Toeplitz type operators, theextension and restriction operators, and calculate the second term in theasymptotic expansion of the optimal constant in the semi-classical version ofOhsawa-Takegoshi extension theorem.

Complex embeddings, Toeplitz operators and transitivity of optimal holomorphic extensions