Canonical bundle formula and degenerating families of volume forms

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.

References in corpus (9)

• The Adjunction Conjecture and its applications
• Gromov-Hausdorff collapsing of Calabi-Yau manifolds
• Hodge metrics and the curvature of higher direct images
• New characterizations of plurisubharmonic functions and positivity of direct image sheaves
• Semi-positivity from Higgs bundles
• On semipositivity theorems
• Singularities and analytic torsion
• Jumping numbers of analytic multiplier ideals (with an appendix by Sébastien Boucksom)
• Restricted volumes on Kähler manifolds

Cited by in corpus (3)

• Special Kähler geometry and holomorphic Lagrangian fibrations
• Boundary asymptotics of the relative Bergman kernel metric for curves
• $L^2$ extension of holomorphic functions for log canonical pairs