Complex analysis and geometry
A note on pseudo-effective vector bundles with vanishing first Chern number over non-Kähler manifolds
Comptes Rendus. Mathématique, Volume 359 (2021) no. 5, pp. 523-531.

In this note, We show that over a compact Hermitian manifold (X,ω) whose metric satisfies ¯ω n-1 = ¯ω n-2 =0, every pseudo-effective vector bundle with vanishing first Chern number is in fact a numerically flat vector bundle.

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DOI: 10.5802/crmath.182
Chen, Yong 1

1 School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P.R. China.
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     title = {A note on pseudo-effective vector bundles with vanishing first {Chern} number over {non-K\"ahler} manifolds},
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Chen, Yong. A note on pseudo-effective vector bundles with vanishing first Chern number over non-Kähler manifolds. Comptes Rendus. Mathématique, Volume 359 (2021) no. 5, pp. 523-531. doi : 10.5802/crmath.182. http://www.numdam.org/articles/10.5802/crmath.182/

[1] Boucksom, Sébastien; Demailly, Jean-Pierre; Păun, Mihao; Peternell, Thomas The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebr. Geom., Volume 22 (2013) no. 2, pp. 201-248 | DOI | Zbl

[2] Demailly, Jean-Pierre Regularization of closed positive currents and Intersection Theory, J. Algebr. Geom., Volume 1 (1992) no. 3, pp. 361-409 | MR | Zbl

[3] Demailly, Jean-Pierre Complex analytic and differential geometry, 2012 (online-book: https://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf)

[4] Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael Compact complex manifolds with numerically effective tangent bundles, J. Algebr. Geom., Volume 3 (1994) no. 2, pp. 295-345 | MR | Zbl

[5] Diverio, Simone Segre forms and Kobayashi–Lübke inequality, Math. Z., Volume 283 (2016) no. 3-4, pp. 1033-1047 | DOI | MR | Zbl

[6] Guler, Dincer On Segre forms of positive vector bundles, Can. Math. Bull., Volume 55 (2012) no. 1, pp. 108-113 | DOI | MR | Zbl

[7] Jost, Jürgen; Yau, Shing-Tung A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry, Acta Math., Volume 170 (1993) no. 2, pp. 221-254 | DOI | MR | Zbl

[8] Li, Chao; Nie, Yanci; Zhang, Xi Numerically flat holomorphic bundles over non-Kähler manifolds (2019) (https://arxiv.org/abs/1901.04680)

[9] Nie, Yanci; Zhang, Xi A note on semistable Higgs bundles over compact Kähler manifolds, Ann. Global Anal. Geom., Volume 48 (2015) no. 4, pp. 345-355 | Zbl

[10] Wu, Xiaojun Pseudo-effective and numerically flat reflexive sheaves (2020) (https://arxiv.org/abs/2004.14676)

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