The asymptotics of the optimal holomorphic extensions of holomorphic jets along submanifolds

TL;DR

This paper derives explicit asymptotic formulas for $L^2$-optimal holomorphic extensions of jets along submanifolds in high tensor powers of positive line bundles, extending previous results from sections to jets.

Contribution

It provides the first explicit asymptotic formula for the extension operator of holomorphic jets in the high tensor power limit, advancing the understanding of extension problems in complex geometry.

Findings

Explicit asymptotic formula for the extension operator

Asymptotic isometry between norms on jets

Extension of previous results from sections to jets

Abstract

We study the asymptotics of the $L^2$-optimal holomorphic extensions of holomorphic jets associated with high tensor powers of a positive line bundle along submanifolds. More precisely, for a fixed complex submanifold in a complex manifold, we consider the operator which for a given holomorphic jet along the submanifold of a positive line bundle associates the $L^2$-optimal holomorphic extension of it to the ambient manifold. When the tensor power of the line bundle tends to infinity, we give an explicit asymptotic formula for this extension operator. This is done by a careful study of the Schwartz kernels of the extension operator and related Bergman projectors. It extends our previous results, done for holomorphic sections instead of jets. As an application, we prove the asymptotic isometry between two natural norms on the space of holomorphic jets: one induced from the ambient manifold and another from the submanifold.