Hostname: page-component-54dcc4c588-wlffp Total loading time: 0 Render date: 2025-09-19T06:17:54.418Z Has data issue: false hasContentIssue false
  • English
  • Français

Holomorphic Vanishing Theorems on Finsler Holomorphic Vector Bundles and Complex Finsler Manifolds

Part of: Partial differential equations on manifolds; differential operators Global differential geometry

Published online by Cambridge University Press:  09 November 2018

Bin Shen
Bin Shen*
Affiliation:
School of Mathematics, Southeast University, 211189, Nanjing, P. R. China Email: shenbin@seu.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we investigate the holomorphic sections of holomorphic Finsler bundles over both compact and non-compact complete complex manifolds. We also inquire into the holomorphic vector fields on compact and non-compact complete complex Finsler manifolds. We get vanishing theorems in each case according to different certain curvature conditions. This work can be considered as generalizations of the classical results on Kähler manifolds and hermitian bundles.

Keywords

complex Finsler metricholomorphic vector fieldholomorphic sectionG-average Ricci curvaturevanishing theorem

MSC classification

Primary: 53C60: Finsler spaces and generalizations (areal metrics)
Secondary: 53C56: Other complex differential geometry 58J60: Relations with special manifold structures (Riemannian, Finsler, etc.)

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 3 , September 2019 , pp. 623 - 641
Copyright
© Canadian Mathematical Society 2018 

Footnotes

The author is supported by the Natural Science Foundation of Jiangsu Province (No. BK20160661) and Zhishan young Scholar Program of SEU (No. 2242019R40055).

References

Abate, M. and Patrizio, G., Finsler Metrics: A Global Approach . Lecture Notes in Math. 1591 , Springer-Verlag, 1994.Google Scholar
Aikou, T., Finsler Geometry on complex vector bundles . In: Riemann–Finsler Geometry , MSRI Publications 50 , 2004, pp. 83–105.Google Scholar
Andreotti, A. and Vesentini, E., Carleman estimates for the Laplace–Beltrami equation on complex manifolds . Inst. Hautes Etudes Sci. Publ. Math. 25(1965), 313–362.Google Scholar
Bochner, S., Vector fields and Ricci curvature . Bull. Amer. Math. Soc. 52(1946), 776–797.Google Scholar
Cao, J. and Wong, P., Finsler geometry of projectivized vector bundles . J. Math. Kyoto Univ. 43(2003), 369–410.Google Scholar
Feng, H., Liu, K., and Wan, X., Chern forms of holomorphic Finsler vector bundles and some applications . Internat. J. Math. 27(2016), 1650030, 22 pp.Google Scholar
Kobayashi, S., Complex Finsler vector bundles . Contemp. Math. 196 , Amer. Math. Soc., Providence, RI, 1996, pp. 133–144.Google Scholar
Kobayashi, S. and Wu, H. H., On holomorphic sections of certain Hermitian vector bundles . Math. Ann. 189(1970), 1–4.Google Scholar
Shen, B., Frankel’s refinement of Bochner’s theorem on forward complete Finsler manifolds . Differential Geom. Appl. 51(2017), 65–75.Google Scholar
Shen, B., Vanishing of Killing vector fields on Finsler manifolds. Kodai Math. J., to appear.Google Scholar
Siu, Y. T., The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds . Ann. Math. 112(1980), 73–111.Google Scholar
Yau, S. T., Some function-theoretic properties of complete Riemannian manifold and their applications to geometry . Indiana Univ. Math. J. 25(1976), 7, 659–670.Google Scholar
Yorozu, S., Holomorphic vector fields on complete Kähler manifolds . Ann. Sci. Kanazawa Univ. 17(1980), 17–21.Google Scholar