On an $L^2$ extension theorem from log-canonical centres with log-canonical measures

TL;DR

This paper investigates an $L^2$ extension theorem from log-canonical centers on compact Kähler manifolds, introducing a holomorphic deformation of norms and providing explicit examples to support the conjectured estimates.

Contribution

It develops a framework for $L^2$ estimates with respect to log-canonical measures and offers explicit computations on projective space to support the theory.

Findings

Holomorphic deformation of $L^2$ norms to lc-measures

Invariant $L^2$ estimates under normalization

Explicit examples on $ extbf{P}^3$ verifying estimates

Abstract

With a view to prove an Ohsawa-Takegoshi type $L^2$ extension theorem with $L^2$ estimates given with respect to the log-canonical (lc) measures, a sequence of measures each supported on lc centres of specific codimension defined via multiplier ideal sheaves, this article is aiming at providing evidence and possible means to prove the $L^2$ estimates on compact K\"ahler manifolds $X$. A holomorphic family of $L^2$ norms on the ambient space $X$ is introduced which is shown to "deform holomorphically" to an $L^2$ norm with respect to an lc-measure. Moreover, the latter norm is shown to be invariant under a certain normalisation which leads to a "non-universal" $L^2$ estimate on compact $X$. Explicit examples on $\mathbb{P}^3$ with detailed computation are presented to verify the expected $L^2$ estimates for extensions from lc centres of various codimensions and to provide hint for the proof of the estimates in general.