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Strongly Pseudo-Effective and Numerically Flat Reflexive Sheaves

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Abstract

In this paper, we discuss the concept of strongly pseudo-effective vector bundle and also introduce strongly pseudo-effective torsion-free sheaves over compact Kähler manifolds. We show that a strongly pseudo-effective reflexive sheaf over a compact Kähler manifold with vanishing first Chern class is in fact a numerically flat vector bundle. A proof is obtained through a natural construction of positive currents representing the Segre classes of strongly pseudo-effective vector bundles.

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Acknowledgements

I thank Jean-Pierre Demailly, my PhD supervisor, for his guidance, patience and generosity. I would like to thank Junyan Cao, Sébastien Boucksom, Simone Diverio, Andreas Höring and Richard Lärkäng for some very useful suggestions on the previous draft of this work. I would also like to express my gratitude to colleagues of Institut Fourier for all the interesting discussions we had. This work is supported by the PhD programme AMX of École Polytechnique and Ministère de l’Enseignement Supérieur et de la Recherche et de l’Innovation, and the European Research Council grant ALKAGE number 670846 managed by J.-P. Demailly. We thank the anonymous reviewer for a very careful reading of this paper, and for insightful comments and suggestions.

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  1. Institut Fourier, Univerisité Grenoble Alpes, 38400, Saint-Martin-d’Hères, France

    Xiaojun Wu

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Wu, X. Strongly Pseudo-Effective and Numerically Flat Reflexive Sheaves. J Geom Anal 32, 124 (2022). https://doi.org/10.1007/s12220-021-00865-0

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