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Complex immersions in Kähler manifolds of positive holomorphic $k$-Ricci curvature

Authors: Fuquan Fang and Sérgio Mendonça
Journal: Trans. Amer. Math. Soc. 357 (2005), 3725-3738
MSC (2000): Primary 32Q15; Secondary 53C55
Published electronically: March 25, 2005
MathSciNet review: 2146646
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Abstract: The main purpose of this paper is to prove several connectedness theorems for complex immersions of closed manifolds in Kähler manifolds with positive holomorphic $k$-Ricci curvature. In particular this generalizes the classical Lefschetz hyperplane section theorem for projective varieties. As an immediate geometric application we prove that a complex immersion of an $n$-dimensional closed manifold in a simply connected closed Kähler $m$-manifold $M$ with positive holomorphic $k$-Ricci curvature is an embedding, provided that $2n\ge m+k$. This assertion for $k=1$ follows from the Fulton-Hansen theorem (1979).

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  • [AF] A. Andteotti; T. Frankel, The Lefschetz theorem on hyperplane sections, Ann. Math 69 (1959), 713-717. MR 0177422 (31:1685)
  • [Ba] W. Barth, Transplating cohomology classes in complex projective space, Amer. J. Math. 92 (1970), 951-967. MR 0287032 (44:4239)
  • [FMR] F. Fang; S. Mendonça; X. Rong, A Connectedness principle in the geometry of positive curvature, preprint, 2002, to appear in Comm. Analysis and Geometry.
  • [Fr] T. Frankel, Manifolds of positive curvature, Pacific J. Math. 11 (1961), 165-174. MR 0123272 (23:A600)
  • [Fu] W. Fulton, On the topology of algebraic varieties, Proc. Symp. in Pure Math. 46 (1987), 15-46. MR 0927947 (89c:14027)
  • [FH] W. Fulton; J. Hansen, A connectedness theorems for projective varieties, with applications to intersections and singularities of mappings, Ann. Math 110 (1979), 159-166. MR 0541334 (82i:14010)
  • [FL] W. Fulton; R. Lazarsfeld, Connectivity and Its Applications in Algebraic Geometry, Lecture Notes in Mathematics 862, Springer-Verlag, 26-92. MR 0644817 (83i:14002)
  • [GK] S. Goldberg; S. Kobayashi, Holomorphic bisectional curvature, J. Differential. Geom. 1 (1967), 225-233. MR 0227901 (37:3485)
  • [GM] M. Goresky; R. MacPherson, Stratified Morse theory, Springer-Verlag, New York, 1988. MR 0932724 (90d:57039)
  • [Gr] A. Gray, Nearly Kähler manifolds, J. Diff. Geom. 4 (1970), 283-309. MR 0267502 (42:2404)
  • [G] P. Griffiths, Hermitian differential geometry, Chern classes, and positive vector bundles, Global Analysis, Univ. Tokyo Press (1969), 195-251. MR 0258070 (41:2717)
  • [GR] K. Grove, Geodesics satisfying general boundary conditions, Comment. Math. Helv. (1973), 376-381. MR 0438386 (55:11300)
  • [Gu] F. F. Guimarães, The integral of the scalar curvature of complete manifolds without conjugate points, J. Differential Geom. 36 (1992), 651-662. MR 1189499 (93j:53055)
  • [KW] M. Kim; J. Wolfson, Theorems of Barth-Lefschetz type on Kähler manifolds of non-negative bisectional curvature, Forum Math. 15 (2003), 261-273.MR 1956967 (2004b:32036)
  • [Le] S. Lefschetz, L'analysis situs et la geometrie algebrique, Gauthier-Villars, Paris (1924). MR 0033557 (11:456c)
  • [Mi] J. Milnor, Morse theory, Ann. Math. Stud. Princeton University Press (1963). MR 0163331 (29:634)
  • [Mo] S. Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), 593-606. MR 0554387 (81j:14010)
  • [MZ] S. Mendonça; D. Zhou, Curvature conditions for immersions of submanifolds and applications, Compositio Math. 137 (2003), 211-226. MR 1985004 (2004c:53093)
  • [Ok] C. Okonek, Barth-Lefschetz theorems for singular spaces, J. Reine. Angew Math. 374 (1987), 24-38. MR 0876219 (88c:14029)
  • [Or] L. Ornea, A theorem on non-negatively curved locally conformal Kaehler manifolds, Rendi. di Matematica 12 (1992), 257-262. MR 1186159 (93h:53071)
  • [SW] R. Schoen; J. Wolfson, Theorems of Barth-Lefschetz types and Morse theory on the spaces of paths, Math. Zeit. 229 (1998), 77-89. MR 1649314 (2000i:58021)
  • [Sh] Z. Shen, On complete manifolds of nonnegativee kth-Ricci curvature, Trans of A.M.S. 338 (1993), 289-310. MR 1112548 (93j:53054)
  • [SY] Y.-T. Siu, S.-T. Yau, Compact Kähler manifolds with positive bisectional curvature, Invent. Math. 59 (1980), 189-204. MR 0577360 (81h:58029)
  • [So] A. Sommese, Complex subspaces of homogeneous complex manifolds II- Homotopy Results, Nagoya Math. J. 86 (1982), 101-129. MR 0661221 (84d:32040)
  • [Wu] H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J 36 (1987), 525-548. MR 0905609 (88k:53068)

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Additional Information

Fuquan Fang
Affiliation: Nankai Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China

Sérgio Mendonça
Affiliation: DepartamentodeAnálise,\hskip1mm Universidade FederalFluminense (UFF), Niterói, 24020-140 RJ Brazil

Received by editor(s): August 5, 2003
Received by editor(s) in revised form: March 10, 2004
Published electronically: March 25, 2005
Additional Notes: The first author was supported by NSFC Grant 19741002, RFDP and the Qiu-Shi Foundation
Article copyright: © Copyright 2005 American Mathematical Society

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