Complex embeddings, Toeplitz operators and transitivity of optimal holomorphic extensions

In a setting of a complex manifold with a 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 L²-norm. We show that for a tower of submanifolds and large tensor powers of the line bundle, the extension operators act transitively modulo some small defect, which is a Toeplitz type operator. We calculate the first significant term in the asymptotic expansion of this “transitivity defect”. As a byproduct, we deduce 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 the extension theorem.