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

TL;DR

This paper investigates the asymptotic behavior of the Ohsawa-Takegoshi extension operator and orthogonal Bergman kernel for high tensor powers of positive line bundles, providing explicit expansions and kernel estimates.

Contribution

It introduces explicit asymptotic expansions for the extension operator and Bergman kernel in the semiclassical limit, with detailed kernel estimates.

Findings

Derived explicit asymptotic expansions for the extension operator.

Established exponential estimates for the Schwartz kernel.

Proved full asymptotic expansion of the kernel in the high tensor power limit.

Abstract

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