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

Fang, Fuquan; Mendonça, Sérgio

doi:10.1090/S0002-9947-05-03675-5

Complex immersions in Kähler manifolds of positive holomorphic $k$-Ricci curvature
HTML articles powered by AMS MathViewer

by Fuquan Fang and Sérgio Mendonça PDF

Trans. Amer. Math. Soc. 357 (2005), 3725-3738 Request permission

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).

References

Aldo Andreotti and Theodore Frankel, The Lefschetz theorem on hyperplane sections, Ann. of Math. (2) 69 (1959), 713–717. MR 177422, DOI 10.2307/1970034
W. Barth, Transplanting cohomology classes in complex-projective space, Amer. J. Math. 92 (1970), 951–967. MR 287032, DOI 10.2307/2373404
F. Fang; S. Mendonça; X. Rong, A Connectedness principle in the geometry of positive curvature, preprint, 2002, to appear in Comm. Analysis and Geometry.
Theodore Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165–174. MR 123272
William Fulton, On the topology of algebraic varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 15–46. MR 927947
William Fulton and Johan Hansen, A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings, Ann. of Math. (2) 110 (1979), no. 1, 159–166. MR 541334, DOI 10.2307/1971249
William Fulton and Robert Lazarsfeld, Connectivity and its applications in algebraic geometry, Algebraic geometry (Chicago, Ill., 1980) Lecture Notes in Math., vol. 862, Springer, Berlin-New York, 1981, pp. 26–92. MR 644817
Samuel I. Goldberg and Shoshichi Kobayashi, Holomorphic bisectional curvature, J. Differential Geometry 1 (1967), 225–233. MR 227901
Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. MR 932724, DOI 10.1007/978-3-642-71714-7
Alfred Gray, Nearly Kähler manifolds, J. Differential Geometry 4 (1970), 283–309. MR 267502
Phillip A. Griffiths, Hermitian differential geometry, Chern classes, and positive vector bundles, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 185–251. MR 0258070
Karsten Grove, Geodesics satisfying general boundary conditions, Comment. Math. Helv. 48 (1973), 376–381. MR 438386, DOI 10.1007/BF02566130
Florêncio F. Guimarães, The integral of the scalar curvature of complete manifolds without conjugate points, J. Differential Geom. 36 (1992), no. 3, 651–662. MR 1189499
Meeyoung Kim and Jon Wolfson, Theorems of Barth-Lefschetz type on Kähler manifolds of non-negative bisectional curvature, Forum Math. 15 (2003), no. 2, 261–273. MR 1956967, DOI 10.1515/form.2003.014
S. Lefschetz, L’analysis situs et la géométrie algébrique, Gauthier-Villars, Paris, 1950 (French). MR 0033557
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
Shigefumi Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), no. 3, 593–606. MR 554387, DOI 10.2307/1971241
Sérgio Mendonça and Detang Zhou, Curvature conditions for immersions of submanifolds and applications, Compositio Math. 137 (2003), no. 2, 211–226. MR 1985004, DOI 10.1023/A:1023932008595
Christian Okonek, Barth-Lefschetz theorems for singular spaces, J. Reine Angew. Math. 374 (1987), 24–38. MR 876219, DOI 10.1515/crll.1987.374.24
L. Ornea, A theorem on non-negatively curved locally conformal Kaehler manifolds, Rend. Mat. Appl. (7) 12 (1992), no. 1, 257–262 (English, with English and Italian summaries). MR 1186159
Richard Schoen and Jon Wolfson, Theorems of Barth-Lefschetz type and Morse theory on the space of paths, Math. Z. 229 (1998), no. 1, 77–89. MR 1649314, DOI 10.1007/PL00004651
Zhong Min Shen, On complete manifolds of nonnegative $k$th-Ricci curvature, Trans. Amer. Math. Soc. 338 (1993), no. 1, 289–310. MR 1112548, DOI 10.1090/S0002-9947-1993-1112548-6
Yum Tong Siu and Shing Tung Yau, Compact Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), no. 2, 189–204. MR 577360, DOI 10.1007/BF01390043
Andrew John Sommese, Complex subspaces of homogeneous complex manifolds. II. Homotopy results, Nagoya Math. J. 86 (1982), 101–129. MR 661221
H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525–548. MR 905609, DOI 10.1512/iumj.1987.36.36029