Extension from lc centres and the extension theorem of Demailly but with estimates

Abstract

This paper generalises the result of Jean-Pierre Demailly on his Ohsawa--Takegoshi-type $L^2$ extension theorem, which guarantees holomorphic extensions for some sections $f$ on analytic subspaces $Y$ defined by multiplier ideal sheaves of plurisubharmonic functions. The result in this paper provides $L^2$ estimates for the extended sections even if those $f$ do not vanish on the singular sets of the reduced loci of the subspaces $Y$, at least for the case when the ambient manifold $X$ is compact and the relevant metrics have only neat analytic singularities. This is achieved by replacing the generalised Ohsawa measure by a measure supported on log-canonical centres.

Some arguments in the proof of Thm. 2.3.3 are erroneous. One of the faulty arguments lies in the estimate on the first line of page 24. The author mistakenly treats the orthogonal decomposition with respect to the unweighted inner product as the one with respect to the weighted one. Contents which are free from irreparable errors are contained in arXiv:1912.08076

References in corpus (4)

• Effectiveness of Demailly's strong openness conjecture and related problems
• Extension of holomorphic functions defined on non reduced analytic subvarieties
• Analytic continuation of residue currents
• A simple proof of the Ohsawa-Takegoshi extension theorem