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On Lelong numbers of generalized Monge–Ampère products

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Abstract

We consider generalized (mixed) Monge–Ampère products of quasiplurisubharmonic functions (with and without analytic singularities) as they were introduced and studied in several articles written by subsets of Andersson, Wulcan, Błocki, Lärkäng, Raufi, Ruppenthal, and the author. We continue these studies and present estimates for the Lelong numbers of pushforwards of such products by proper holomorphic submersions. Furthermore, we apply these estimates to Chern and Segre currents of pseudoeffective vector bundles. Among other corollaries, we obtain the following generalization of a recent result by Wu. If the non-nef locus of a pseudoeffective vector bundle E on a Kähler manifold is contained in a countable union of k-codimensional analytic sets, and if the k-power of the first Chern class of E is trivial, then E is nef.

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Notes

  1. Following the notation usual for currents, we call a (1, 1)-form \(\alpha \) positive if \(\alpha \ge 0\).

  2. To be more precise, on all V from before.

  3. As defined by [7, Def. 3.2]; also called singularly (semi-) positive in the sense of Griffiths.

  4. A singular Finsler metric h on \(E^*\) is given by \(E^*\rightarrow [0,\infty ], (p,\xi )\mapsto \Vert \xi \Vert _{h(p)}\) with \(\Vert \lambda \xi \Vert _{h(p)}=|\lambda |{\cdot }\Vert \xi \Vert _{h(p)}\).

  5. In the literature, sometimes called weakly to distinguish it from strongly pseudoeffective.

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Acknowledgements

The author is very grateful to Elizabeth Wulcan and the unknown referee for their valuable comments and suggestions which helped to improve the article a lot. The author was supported by JSPS KAKENHI Grant number JP20K14319.

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  1. Faculty of Engineering, Kyoto University of Advanced Science, Kyoto, 615-8577, Japan

    Martin Sera

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Correspondence to Martin Sera.

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In memory of Jean-Pierre Demailly.

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Sera, M. On Lelong numbers of generalized Monge–Ampère products. Math. Z. 304, 46 (2023). https://doi.org/10.1007/s00209-023-03284-9

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