Complex embeddings, Toeplitz operators and transitivity of optimal holomorphic extensions

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