Canonical bundle formula and degenerating families of volume forms

TL;DR

This paper provides an analytic characterization of the canonical bundle formula using fiberwise integration, confirming a folklore conjecture and connecting it to $L^2$ metrics and singularity analysis.

Contribution

It introduces an analytic perspective on the canonical bundle formula, utilizing $L^2$ metrics and valuative equivalence, and applies this to singularity analysis and semipositivity questions.

Findings

Confirmed a folklore conjecture on the analytic characterization of the canonical bundle formula.

Identified the singularity of the Ohsawa measure in $L^2$ extension theorems.

Provided a partial answer to Berndtsson's question on semipositivity.

Abstract

Canonical bundle formula due to Kawamata and others has played fundamental roles in algebraic geometry. We show that the canonical bundle formula has analytic characterization in terms of fiberwise integration, which confirms a folklore conjecture. The proof uses $L^2$ metrics and the valuative equivalence of plurisubharmonic singularities. As an application, we identify the singularity of the Ohsawa measure in a general $L^2$ extension theorem of Demailly for log canonical pairs. As another consequence, we give a partial answer to a question of Berndtsson on semipositivity theorems.