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...

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Bibliographic Details
Published inInventiones mathematicae Vol. 202; no. 3; pp. 1167 - 1198
Main Authors Collins, Tristan C., Tosatti, Valentino
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2015
Springer Nature B.V
Subjects
Loci
Manifolds
Mathematics
Mathematics and Statistics
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Summary: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.
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-015-0585-9