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 $\left(X,\omega \right)$ whose metric satisfies $\partial \overline{\partial }{\omega }^{n-1}=\partial \overline{\partial }{\omega }^{n-2}=0$, every pseudo-effective vector bundle with vanishing first Chern number is in fact a numerically flat vector bundle.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.182
Yong Chen 1

1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P.R. China.
@article{CRMATH_2021__359_5_523_0,
author = {Yong Chen},
title = {A note on pseudo-effective vector bundles with vanishing first {Chern} number over {non-K\"ahler} manifolds},
journal = {Comptes Rendus. Math\'ematique},
pages = {523--531},
publisher = {Acad\'emie des sciences, Paris},
volume = {359},
number = {5},
year = {2021},
doi = {10.5802/crmath.182},
zbl = {07371712},
language = {en},
}
TY  - JOUR
AU  - Yong Chen
TI  - A note on pseudo-effective vector bundles with vanishing first Chern number over non-Kähler manifolds
JO  - Comptes Rendus. Mathématique
PY  - 2021
DA  - 2021///
SP  - 523
EP  - 531
VL  - 359
IS  - 5
PB  - Académie des sciences, Paris
UR  - https://zbmath.org/?q=an%3A07371712
UR  - https://doi.org/10.5802/crmath.182
DO  - 10.5802/crmath.182
LA  - en
ID  - CRMATH_2021__359_5_523_0
ER  - 
%0 Journal Article
%A Yong Chen
%T A note on pseudo-effective vector bundles with vanishing first Chern number over non-Kähler manifolds
%J Comptes Rendus. Mathématique
%D 2021
%P 523-531
%V 359
%N 5
%U https://doi.org/10.5802/crmath.182
%R 10.5802/crmath.182
%G en
%F CRMATH_2021__359_5_523_0
Yong Chen. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.182/

[1] Sébastien Boucksom; Jean-Pierre Demailly; Mihao Păun; Thomas Peternell 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 | Article | Zbl 1267.32017

[2] Jean-Pierre Demailly Regularization of closed positive currents and Intersection Theory, J. Algebr. Geom., Volume 1 (1992) no. 3, pp. 361-409 | MR 1158622 | Zbl 0777.32016

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

[4] Jean-Pierre Demailly; Thomas Peternell; Michael Schneider Compact complex manifolds with numerically effective tangent bundles, J. Algebr. Geom., Volume 3 (1994) no. 2, pp. 295-345 | MR 1257325 | Zbl 0827.14027

[5] Simone Diverio Segre forms and Kobayashi–Lübke inequality, Math. Z., Volume 283 (2016) no. 3-4, pp. 1033-1047 | Article | MR 3519994 | Zbl 1347.53022

[6] Dincer Guler On Segre forms of positive vector bundles, Can. Math. Bull., Volume 55 (2012) no. 1, pp. 108-113 | Article | MR 2932990 | Zbl 1239.53092

[7] Jürgen Jost; Shing-Tung Yau 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 | Article | MR 1226528 | Zbl 0806.53064

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

[9] Yanci Nie; Xi Zhang A note on semistable Higgs bundles over compact Kähler manifolds, Ann. Global Anal. Geom., Volume 48 (2015) no. 4, pp. 345-355 | Zbl 1331.53038

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

Cited by Sources: