Kähler currents and null loci

Abstract

We prove that the non-Kähler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kähler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein-Lazarsfeld-Mustaţȃ-Nakamaye-Popa. As an application, we show that finite time non-collapsing singularities of the Kähler-Ricci flow on compact Kähler 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 Kähler metrics on Calabi-Yau manifolds to the nonalgebraic case.

The Calabi–Yau theorem and Kähler currents

“…The main theorem of [10], which reproves and generalizes algebro-geometric results by Nakamaye [32] and Ein-Lazarsfeld-Mustaţȃ-Nakamaye-Popa [18], then can be stated as follows: Theorem 2.5 (Collins-Tosatti [10]). Let X be a compact complex manifold and α a closed real (1, 1) form whose cohomology class [α] is nef and big.…”

“…Constructing such currents is of great importance for applications to algebraic geometry, see e.g. [3,5,10,13,16] and references therein. Very recently a new point of view on the mass concentration technique was discovered by Chiose [8], which greatly simplifies the picture.…”

“…We also briefly explain how Theorem 2.3 together a recent theorem of Collins and the author [10] provides another proof of Demailly-Pȃun's Nakai-Moishezon criterion for Kähler manifolds [16], and we use this to characterize the points where the Seshadri constant of a nef class vanishes.…”

The complex Monge–Ampère type equation on compact Hermitian manifolds and applications

“…On such a manifold there is a singular set which plays an important role in studying the regularity of the flow (4.14). It is defined in [CoTo13].…”

“…Gill has obtained the following E is the smallest set that we can take because of a result of Collins and Tosatti [CoTo13]. It is a generalisation of [DP04, Theorem 0.5] to manifolds in the Fujiki class.…”