Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-11T13:21:44.072Z Has data issue: false hasContentIssue false
  • English
  • Français

Complex subspaces of homogeneous complex manifolds II—Homotopy Results

Published online by Cambridge University Press:  22 January 2016

Andrew John Sommese
Andrew John Sommese*
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana, 46556, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Lefschetz hyperplane section theorem has roots going back at least to Picard, but it was Lefschetz [20] who first stated and proved it in the modern form for integer homology. Later it was improved up to the homotopy level by Andreotti-Frankel [1] and Bott [8] using an idea of Thorn. Numerous generalizations along the same lines have appeared, e.g. [14, Theorem H], [19], [24, App. II] etc.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 86 , June 1982 , pp. 101 - 129
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1982

References

[ 1 ] Andreotti, A. and Frankel, T., The Lefschetz theorem on hyperplane sections, Ann. of Math., 69 (1959), 713–717.Google Scholar
[ 2 ] Andreotti, A. and Grauert, H., Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 90 (1962), 193–259.Google Scholar
[ 3 ] Barth, W., Der Abstand von einer algebraischen Mannigfaltigkeit im komplex-projectiven Raum, Math. Annalen, 187 (1970), 150–162.Google Scholar
[ 4 ] Barth, W., Transplanting cohomology classes in complex-projective space, Amer. Jour. of Math., 92 (1970), 951–967.Google Scholar
[ 5 ] Barth, W., Larsen’s theorem on the homotopy groups of projective manifolds of small embedding codimension, Proc. Symp. in Pure Math., 29 (1975), 307–313.CrossRefGoogle Scholar
[ 6 ] Barth, W. and Larsen, M. E., On the homotopy groups of complex projective manifolds, Math. Scand., 30 (1972), 88–94.CrossRefGoogle Scholar
[ 7 ] Barth, W. and van de Ven, A., A decomposability criterion for algebraic 2 bundles on projective space, Invent. Math., 25 (1974), 91–106.Google Scholar
[ 8 ] Bott, R., On a theorem of Lefschetz, Mich. Math. J., 6 (1959), 211–216.Google Scholar
[ 9 ] Chern, S. S., Complex Manifolds without Potential Theory, Math. Studies 15, D. Van Nostrand Co., Princeton, 1967.Google Scholar
[10] Dold, A. and Thorn, R., Quasifaserungen und unendlich Symmetrische Produkte, Ann. of Math., 67 (1958), 239–273.Google Scholar
[11] Fischer, G., Complex Analytic Geometry, Lecture Notes in Math. 538, Springer Verlag, New York, 1976.Google Scholar
[12] Fritzsche, K., q-konvex Restmengen in kompakten komplexen Mannigfaltigkeiten, Math. Annalen, 221 (1976), 251–273.Google Scholar
[13] Grauert, H. and Mülich, G., Vektorbündel vom Rang 2 über dem n-dimensional komplex-projektiven Raum, Manuscripta Math., 16 (1975), 75–100.Google Scholar
[14] Griffiths, P. A., Hermitian differential geometry, Chern classes, and positive vector bundles, in: Global Analysis, Papers in honor of K. Kodaira, 185–251, Princeton University Press, 1969.Google Scholar
[15] Hartshorne, R., Cohomology of non-complete algebraic varieties, Compos. Math., 23 (1971), 257–264.Google Scholar
[16] Hartshorne, R., Varieties of small codimension in projective space, Bull. Amer. Math. Soc, 80 (1974), 1017–1032.CrossRefGoogle Scholar
[17] Hartshorne, R. and Speiser, R., Cohomological dimension in characteristic p , Ann. of Math., (2) 105 (1977), 45–79.CrossRefGoogle Scholar
[18] Hodge, W. V. D. and Pedoe, E., Methods of Algebraic Geometry, Vol. 2, Cambridge Univ. Press, 1952.Google Scholar
[19] Kaup, L. and Weidner, H., Mayer-Vietoris Sequenzen und Lefschetzsatze für mehrfache Hyperflächenschnitte in der Homotopie, Math. Zeit., 142 (1975), 243–269.Google Scholar
[20] Lefschetz, S., L’Analysis situs et la géométrie algébrique, Selected Papers by Lefschetz, S., Chelsea Publishing Company, New York, 1971.Google Scholar
[21] Matsushima, Y., Heisenberg groups and holomorphic vector bundles over a complex torus, Nagoya Math. J., 61 (1976), 161–195.Google Scholar
[22] Milnor, J., Morse Theory, Princeton University Press, 1963, Second printing, 1965.CrossRefGoogle Scholar
[23] Ogus, A., Local cohomological dimension of algebraic varieties, Ann. Math., 98 (1973), 327–365.CrossRefGoogle Scholar
[24] Sommese, A. J., On manifolds that cannot be ample divisors, Math. Ann., 221 (1976), 55–72.Google Scholar
[25] Sommese, A. J., Theorems of Barth-Lefschetz type for complex subspaces of homogeneous complex manifolds, Proc. Nat. Acad. Sci. USA, 74 (1977), 1332–1333.Google Scholar
[26] Sommese, A. J., Submanifolds of Abelian varieties, Math. Ann., 233, (1978), 229–256.Google Scholar
[27] Sommese, A. J., Concavity theorems, Math. Ann., 235 (1978), 37–53.Google Scholar
[28] Sommese, A. J., Complex Subspaces of Homogeneous Complex Manifolds, I—Transplanting Theorems, Duke Jour, of Math., 46 (1979), 527–548.Google Scholar
[29] Sommese, A. J., A convexity theorem, to appear in Proc. Symp. in Pure Math, on Singularities, Arcata, 1981.Google Scholar
[30] Spanier, E. H., Algebraic Topology, New York, McGraw-Hill, 1966.Google Scholar