Complex embeddings, Toeplitz operators and transitivity of optimal holomorphic extensions

TL;DR

This paper investigates the asymptotic behavior of optimal holomorphic extension operators in complex geometry, revealing transitivity properties and Toeplitz operator relations in the semiclassical limit, with precise expansion calculations.

Contribution

It introduces a novel analysis of the transitivity of extension operators and their connection to Toeplitz operators in the semiclassical setting, providing new asymptotic expansion results.

Findings

Transitivity of extension operators holds modulo small Toeplitz-type defects.

Calculated the leading term of the transitivity defect's asymptotic expansion.

Derived composition rules and second-term expansion of the extension constant.

Abstract

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.