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

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 Kähler 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.

22 pages; v3: minor typos fixed; published online in Math. Z., view-only version via https://rdcu.be/cFDPA

References in corpus (3)

• Analytic continuation of residue currents
• Extension with log-canonical measures and an improvement to the plt extension of Demailly--Hacon--Paun
• On the Ohsawa-Takegoshi extension theorem

Cited by in corpus (2)

• A new definition of analytic adjoint ideal sheaves via the residue functions of log-canonical measures I
• Residue functions and Extension problems