Geodesic
Holomorphic Vanishing Theorems on Finsler Holomorphic Vector Bundles and Complex Finsler Manifolds
Canadian mathematical bulletin, Tome 62 (2019) no. 3, pp. 623-641

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/S0008439518000127
Mots-clés : complex Finsler metric, holomorphic vector field, holomorphic section, G-average Ricci curvature, vanishing theorem 
Shen, Bin. Holomorphic Vanishing Theorems on Finsler Holomorphic Vector Bundles and Complex Finsler Manifolds. Canadian mathematical bulletin, Tome 62 (2019) no. 3, pp. 623-641. doi: 10.4153/S0008439518000127
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     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439518000127/}
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[1] Abate, M. and Patrizio, G., Finsler Metrics: A Global Approach . Lecture Notes in Math. 1591 , Springer-Verlag, 1994. Google Scholar

[2] Aikou, T., Finsler Geometry on complex vector bundles . In: Riemann–Finsler Geometry , MSRI Publications 50 , 2004, pp. 83–105. Google Scholar

[3] 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

[4] Bochner, S., Vector fields and Ricci curvature . Bull. Amer. Math. Soc. 52(1946), 776–797. Google Scholar

[5] Cao, J. and Wong, P., Finsler geometry of projectivized vector bundles . J. Math. Kyoto Univ. 43(2003), 369–410. Google Scholar

[6] 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

[7] Kobayashi, S., Complex Finsler vector bundles . Contemp. Math. 196 , Amer. Math. Soc., Providence, RI, 1996, pp. 133–144. Google Scholar

[8] Kobayashi, S. and Wu, H. H., On holomorphic sections of certain Hermitian vector bundles . Math. Ann. 189(1970), 1–4. Google Scholar

[9] Shen, B., Frankel’s refinement of Bochner’s theorem on forward complete Finsler manifolds . Differential Geom. Appl. 51(2017), 65–75. Google Scholar

[10] Shen, B., Vanishing of Killing vector fields on Finsler manifolds. Kodai Math. J., to appear. Google Scholar

[11] 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

[12] 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

[13] Yorozu, S., Holomorphic vector fields on complete Kähler manifolds . Ann. Sci. Kanazawa Univ. 17(1980), 17–21. Google Scholar

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