Kähler currents and null loci

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–Mustaţă–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.