A note on pseudo-effective vector bundles with vanishing first Chern number over non-Kähler manifolds

Yong Chen

doi:10.5802/crmath.182

Complex analysis and geometry

A note on pseudo-effective vector bundles with vanishing first Chern number over non-Kähler manifolds

Yong Chen ¹

¹ School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P.R. China.

Comptes Rendus. Mathématique, Volume 359 (2021) no. 5, pp. 523-531.

Abstract

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

Received: 2020-10-25
Revised: 2020-12-19
Accepted: 2021-01-20
Published online: 2021-07-13

Zbl

DOI: 10.5802/crmath.182

Author's affiliations:

Yong Chen ¹

¹ School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P.R. China.

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     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},
}

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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/

References
Cited by

[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 | DOI | Zbl

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

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

[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 | Zbl

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

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

[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 | DOI | MR | Zbl

[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

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

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