$L^2$ extension of holomorphic functions for log canonical pairs

Abstract

In a general L 2 extension theorem of Demailly for log canonical pairs, the L 2 condition with respect to the Ohsawa measure determines when a function can be extended. We give a geometric characterization of this analytic condition in terms of log canonical centers. Moreover, the singularity of the Ohsawa measure is shown to be determined according to subadjunction on each maximal log canonical center with a unique log canonical place. This implies that the L 2 condition is essentially equivalent to one that appeared in a previous L 2 extension theorem of the author, which enables to combine these results into common generalization/strengthening. Contents 1. Introduction 1 2. Log canonical centers 8 3. Proofs of the main results 13 4. L 2 extension theorem for a log canonical center 20 5. Kawamata metric and Ohsawa measure 22 References 26Key words and phrases. L 2 extension theorems; Ohsawa-Takegoshi extension; Log canonical center; Ohsawa measure; Subadjunction.1 More precisely, Ψ is allowed to be 'log canonical along Y ' in [O5, Thm.4] as well as in [D15, Thm. 2.8]. See Remark 1.3. 2 However the formulation of [K07, Thm. 4.2] did not use Ψ functions. Also it dealt with divisorial pairs only but the arguments apply to more general pairs as in this paper. In this paper, we will assume X smooth in [K07, Thm. 4.2]. 3 This condition should be made explicit in [K07, Thm. 4.2].