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

TL;DR

This paper investigates the limitations of the $L^2$ extension theorem for holomorphic functions, especially in cases with non-analytic singularities like toric ones, revealing instances where the extension fails despite having a unique log canonical place.

Contribution

It demonstrates that for certain non-analytic singularities, the $L^2$ extension theorem does not apply, highlighting limitations in the current understanding of extension problems.

Findings

Existence of defining functions with only zero function having finite Ohsawa norm.

Counterexamples among simple non-analytic singularities such as toric singularities.

The $L^2$ extension theorem may be void in cases with certain non-analytic singularities.

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.