K\"ahler currents and null loci

TL;DR

This paper establishes a link between the non-Kahler locus and null locus on certain complex manifolds, providing new insights into Kahler-Ricci flow singularities and extending Ricci-flat metric degeneration results.

Contribution

It proves the equality of non-Kahler and null loci for nef and big classes, offering an analytic proof of a key algebraic geometry theorem, and applies this to Kahler-Ricci flow singularities.

Findings

Non-Kahler locus equals null locus on bimeromorphic Kahler manifolds.

Finite time singularities of Kahler-Ricci flow occur along analytic subvarieties.

Extension of Ricci-flat metric degeneration results to nonalgebraic Calabi-Yau manifolds.

Abstract

We prove that the non-Kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kahler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein-Lazarsfeld-Mustata-Nakamaye-Popa. As an application, we show that finite time non-collapsing singularities of the Kahler-Ricci flow on compact Kahler manifolds always form along analytic subvarieties, thus answering a question of Feldman-Ilmanen-Knopf and Campana. We also extend the second author's results about noncollapsing degenerations of Ricci-flat Kahler metrics on Calabi-Yau manifolds to the nonalgebraic case.