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

TL;DR

The paper investigates L^2 extension theorems formulated through the Ohsawa norm $||\cdot||_\Psi$ for quasi-psh defining functions with log canonical and non-analytic singularities. It introduces toric plurisubharmonic functions and translates finiteness of the Ohsawa norm into convex-analytic criteria on the Newton convex body $P$ and its polar $P^\circ$, leveraging monomial valuations to connect analytic singularities with Newton geometry. The main results show that toric $\Psi$ can have a unique log canonical place yet yield an entirely singular Ohsawa norm on the relevant locus, voiding the extension, and provide a precise criterion: $||\cdot||_\Psi$ is not singular at the origin if and only if the boundary of $P$ is locally a hyperplane near $\,(1,\dots,1)$. These findings reveal intrinsic obstructions to universal L^2 extension in non-analytic settings and establish a deep link between analytic singularities and convex geometry that guides how to construct or obstruct extensions.

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.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 2.4
• proof
• Lemma 2.5