$L^2$ extension of holomorphic functions and log canonical places

Dano Kim; Xu Wang

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arXiv · 2505.00801

$L^2$ extension of holomorphic functions and log canonical places

Dano Kim·Xu Wang

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Abstract

In an influential $L^2$ extension theorem due to Demailly, the finiteness of an $L^2$ norm called the Ohsawa norm determines whether a given holomorphic function can be extended. This result has been further generalized by Zhou and Zhu to the case when the quasi-plurisubharmonic defining function of the subvariety has non-analytic singularities. We show that, however, there exist many instances of such defining functions for which only the zero function has finite Ohsawa norm, so that the $L^2$ extension statement is void in such cases, even when it has a unique log canonical place. Such a defining function occurs already among some of the simplest non-analytic singularities, namely toric ones.

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Dano Kim, Xu Wang. 2025-05-01. $L^2$ extension of holomorphic functions and log canonical places. https://arxiv.org/abs/2505.00801

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