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Kähler currents and null loci

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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–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.

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Acknowledgments

We would like to thank D.H. Phong for all his advice and support, and S. Boucksom, S. Dinew, G. Székelyhidi, B. Weinkove, S. Zelditch, A. Zeriahi, Y. Zhang and Z. Zhang for comments and discussions. We also thank the referees for carefully reading our manuscript, for helpful comments and corrections, and for a useful suggestion which simplified our original proof of Theorem 3.2.

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Authors and Affiliations

  1. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA, 02138, USA

    Tristan C. Collins

  2. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60201, USA

    Valentino Tosatti

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  1. Tristan C. Collins
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  2. Valentino Tosatti
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Correspondence to Valentino Tosatti.

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Dedicated to D.H. Phong with admiration on the occasion of his 60th birthday

The second-named author was supported in part by a Sloan Research Fellowship and NSF Grants DMS-1236969 and DMS-1308988.

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Collins, T.C., Tosatti, V. Kähler currents and null loci. Invent. math. 202, 1167–1198 (2015). https://doi.org/10.1007/s00222-015-0585-9

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