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

TL;DR

This paper extends the $L^2$ extension theorem for holomorphic functions on log canonical pairs by providing a geometric characterization of the analytic extension criterion using algebraic geometry, addressing key questions about the Ohsawa measure.

Contribution

It introduces a geometric perspective to the $L^2$ extension criterion for log canonical pairs, linking analytic conditions to algebraic geometry concepts.

Findings

Characterization of when the Ohsawa measure lacks a smooth positive density.

A geometric interpretation of the $L^2$ extension criterion.

Resolution of an open question about the nature of the Ohsawa measure.

Abstract

In a general $L^2$ extension theorem of Demailly for log canonical pairs, the $L^2$ criterion with respect to a measure called the Ohsawa measure determines when a given holomorphic function can be extended. Despite the analytic nature of the Ohsawa measure, we establish a geometric characterization of this analytic criterion using the theory of log canonical centers from algebraic geometry. Along the way, we characterize when the Ohsawa measure fails to have generically smooth positive density, which answers an essential question arising from Demailly's work.