Abstract
A horizontal Hodge Laplacian operator ▭ H is defined for Hermitian holomorphic vector bundles over PTM on Kähler Finsler manifold, and the expression of ▭ H is obtained explicitly in terms of horizontal covariant derivatives of the Chern-Finsler connection. The vanishing theorem is obtained by using the \(\partial _\mathcal{H} \bar \partial _\mathcal{H} \)-method on Kähler Finsler manifolds.
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References
Bochner, S., Vector fields and Ricci curvature, Bull. Amer. Math. Soc., 52, 1946, 776–797.
Bochner, S., Curvature in Hermitian metric, Bull. Amer. Math. Soc., 53, 1947, 179–195.
Bochner, S., Curvature and Betti numbers, I, II, Ann. of Math., 49, 1948, 379–390; 50, 1949, 77–93.
Yano, K. and Bochner, S., Curvature and Betti Numbers, Princeton Univ. Press, Princeton, 1953.
Wu, H. H., The Bochner Technique in Differential Geometry, Harwood Academic Publishers, London, Paris, 1988.
Morrow, J. and Kodaira, K., Complex Manifold, Holt, Rinehart and Winston, New York, 1971.
Griffiths, P. and Harris, J., Principles of Algebraic Geometry, John Wiley and Sons Inc., New York, 1978.
Kobayashi, S., Differential Geometry of Complex Vector Bundles, Princeton Univ. Press, New Jersey, 1987.
Kodaira K., On a differential-geometric method in the theory of analytic stacks, Proc. Nat. Acad. Sci., 39, 1953, 1268–1273.
Siu, Y. T., The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math., 112, 1980, 73–111.
Siu, Y. T., Complex analyticity of harmonic maps, vanishing theorems and Lefchetz theorems, J. Diff. Geom., 17, 1982, 55–138.
Bao, D., Chern, S. S. and Shen, Z., An Introducetion to Riemann-Finsler Geometry, Springer-Verlag, New York, 2000.
Chern, S. S., Finsler geometry is just Riemannian geometry without the quadratic restriction, AMS Notices, 43(9), 1996, 959–963.
Bao, D., Chern, S. S. and Shen, Z., Finsler Geometry (Proceedings of the Joint Summer Research Conference on Finsler Geometry, July 16–20, 1995, Seattle, Washington), Cont. Math., Vol. 196, A. M. S., Providence, RI, 1996.
Shen, Z., Lectures on Finsler Geometry, World Scientific Publishers, Singapore, 2001.
Abate, M. and Patrizio, G., Finsler metric-A global approach, Lecture Notes in Math, Vol. 1591, Springer- Verlag, Berlin, 1994.
Wong, P. M., A survey of complex Finsler geometry, Adv. Stud. Pure Math., 48, 2007, 375–433.
Chen, B., Shen, Y., Kähler Finsler metric are actually strongly Kähler, Chin. Ann. Math., 30B(2), 2009, 173–178.
Bao, D. and Lackey, B., A Hodge decomposition theorem for Finsler spaces, C. R. Acad. Sci. Paris, 323(1), 1996, 51–56.
Antonelli, P. and Bao, D., Finslerian Laplacians and Applications, MAIA 459, Kluwer Academic Publishers, Dordrecht, 1998.
Zhong, C. and Zhong, T., Horizontal α-Laplacian on complex Finsler manifolds, Science in China, Ser. A, 48(supp.), 2005, 377–391.
Zhong, C. and Zhong, T., Hodge decomposition theorem on strongly Kähler Finsler manifolds, Science in China, Ser. A, 49(11), 2006, 1696–1714.
Zhong, C., Laplacians on the holomorphic tangent bundles of a Kähler manifolds, Science in China, Ser. A, 52(12), 2009, 2841–2854.
Aikou, T., On complex Finsler manifolds, Rep. Fac. Kagoshima Univ., 35, 1991, 9–25.
Xiao, J., Zhong, T. and Qiu, C., Bochner technique in Strongly Kähler Finsler manifolds, Acta Math. Sci., 30B(1), 2010, 89–106.
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This work was supported by the National Natural Science Foundation of China (No. 11712777) and the Scientific Research Foundation of Shanghai University of Engineering Science (No. E1-0501-14-0112).
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Xiao, J., Qiu, C. & Zhong, T. Bochner-Kodaira techniques on Kähler Finsler manifolds. Chin. Ann. Math. Ser. B 36, 125–140 (2015). https://doi.org/10.1007/s11401-014-0871-7
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DOI: https://doi.org/10.1007/s11401-014-0871-7