Extension of Jets With $L^2$ Estimates, and an Application

TL;DR

This paper develops an improved $L^2$ extension theorem for normal jets from hypersurfaces, with a focus on controlling the growth of the extension constant, and applies it to extend positively curved metrics in complex geometry.

Contribution

It introduces a new $L^2$ extension theorem with a universal constant depending on the jet order, combining classical and modern approaches, and applies it to metric extension problems.

Findings

Established an extension theorem with a constant $C^k$ where $C$ is universal.

Extended positively curved singular Hermitian metrics from hypersurfaces.

Unified classical and new $L^2$ extension methods.

Abstract

We study the problem of extension of normal jets from a hypersurface, with focus on the growth order of the constant. Using aspects of the standard, twisted approach for $L^2$ extension and of the new approach to $L^2$ extension introduced by Berndtsson and Lempert, we are able to obtain an extension theorem with a constant $C^k$ where $C$ is universal and $k$ is the jet order. We then use the jet extension theorem to extend positively curved singular Hermitian metrics from smooth, deformably pseudoeffective hypersurfaces in projective manifolds.