Abstract
We study a decomposition of a holomorphic vector bundle with connection which need not be endowed with any metrics, which is a generalization of an orthogonal decomposition of a Hermitian holomorphic vector bundle. We first derive several results on the induced connections, the second fundamental forms of subbundles and curvature forms of the connections. We next apply these results to a complex affine immersion. Especially, we give elementary self-contained proofs of the fundamental theorems for a complex affine immersion to a complex affine space.
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Abe, N. and Hasegawa, K.: An affine submersion with horizontral distribution and its applications, Differential Geom. Appl., 14 (2001), 235–250.
Atiyah, M. F.: Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., 85 (1957), 181–207.
Bott, R. and Chern, S. S.: Hermitian vector bundles and the equidistribution of the zeros of their holomorphic sections, Acta. Math., 114 (1965), 71–112.
Dillen, F.: The affine differential geometry of complex hyper surfaces, Med. Konink. Acad. Wetensch. Belgie, 52 (1990), 89–112.
Dillen, F. and Vranken, L.: Complex affine hypersurfaces of ℂn+1, part1, Bull. Soc. Math. Belg. Sér. B, 40 (1988), 245–271.
Furuhata, H. and Matsuzoe, H.: Holom,orphic centroaffine immersions and the Lelieuvre correspondence, Results Math., 33 (1998), 294–305.
Hasegawa, K.: The fundamental theorems for affine immersions into hyperquadrics and its applications, Monatsh. Math., 131 (2000), 37–48.
Kobayashi, S.: Differential Geometry of Complex Vector Bundles, Iwanami Shoten, Publishers and Princeton University Press, 1987.
Nomizu, K., Pinkall, U. and Podesta, F.: On the geometry of affine Kahler immersions, Nagoya Math. J., 120 (1990), 205–222.
Okuda, T.: On the fundamental theorems for purely real immersions, Master thesis 1999, SUT.
Opozda, B.: Fundamental thorem,s for complex affine hypersurfaces, Kobe J. Math., 10 (1993), 133–146.
Opozda, B.: Equivalence theorems for complex affine hypersurfaces, Results Math., 27 (1995), 316–327.
Opozda, B.: On some properties of the curvature and Ricci tensor in complex affine geometry, Geom. Dedicata, 55 (1995), 141–163.
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Abe, N., Kurosu, S. A decomposition of a holomorphic vector bundle with connection and its applications to complex affine immersions. Results. Math. 44, 3–24 (2003). https://doi.org/10.1007/BF03322907
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DOI: https://doi.org/10.1007/BF03322907