Homotopical variations and high-dimensional Zariski-van Kampen theorems. (English) Zbl 1086.14015
The authors set out to generalize the Zariski-van Kampen theorem to higher dimensions. This includes the computations of higher homotopy groups of quasi-projective complex varieties using hyperplane pencils and homotopical variation operators.
Reviewer: Alexandru Dimca (Nice)
MSC:
| 14F35 | Homotopy theory and fundamental groups in algebraic geometry |
| 14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |
| 32S50 | Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants |
References:
| [1] | V. I. Arnol\(^{\prime}\)d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from the Russian by Hugh Porteous; Translation revised by the authors and James Montaldi. |
| [2] | D. Cheniot, Une démonstration du théorème de Zariski sur les sections hyperplanes d’une hypersurface projective et du théorème de Van Kampen sur le groupe fondamental du complémentaire d’une courbe projective plane, Compositio Math. 27 (1973), 141 – 158 (French). · Zbl 0294.14010 |
| [3] | Denis Chéniot, Topologie du complémentaire d’un ensemble algébrique projectif, Enseign. Math. (2) 37 (1991), no. 3-4, 293 – 402 (French). · Zbl 0762.55003 |
| [4] | D. Chéniot, Vanishing cycles in a pencil of hyperplane sections of a non-singular quasi-projective variety, Proc. London Math. Soc. (3) 72 (1996), no. 3, 515 – 544. · Zbl 0851.14003 · doi:10.1112/plms/s3-72.3.515 |
| [5] | D. Chéniot and A. Libgober, Zariski-van Kampen theorem for higher-homotopy groups, J. Inst. Math. Jussieu 2 (2003), no. 4, 495 – 527. · Zbl 1081.14505 · doi:10.1017/S1474748003000148 |
| [6] | Mark Goresky and Robert MacPherson, Stratified Morse theory, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, R.I., 1983, pp. 517 – 533. · Zbl 0526.57022 |
| [7] | 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. · Zbl 0639.14012 |
| [8] | Helmut A. Hamm and Lê Dũng Tráng, Lefschetz theorems on quasiprojective varieties, Bull. Soc. Math. France 113 (1985), no. 2, 123 – 142 (English, with French summary). · Zbl 0602.14009 |
| [9] | P. J. Hilton and S. Wylie, Homology theory: An introduction to algebraic topology, Cambridge University Press, New York, 1960. · Zbl 0091.36306 |
| [10] | Lê Dũng Tráng and B. Teissier, Cycles evanescents, sections planes et conditions de Whitney. II, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 65 – 103 (French, with English summary). · Zbl 0532.32003 |
| [11] | A. Libgober, Homotopy groups of the complements to singular hypersurfaces. II, Ann. of Math. (2) 139 (1994), no. 1, 117 – 144. · Zbl 0815.57017 · doi:10.2307/2946629 |
| [12] | William S. Massey, A basic course in algebraic topology, Graduate Texts in Mathematics, vol. 127, Springer-Verlag, New York, 1991. · Zbl 0725.55001 |
| [13] | Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. · Zbl 0145.43303 |
| [14] | Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. · Zbl 0054.07103 |
| [15] | E.R. van Kampen, “On the fundamental group of an algebraic curve”, Amer. J. Math. 55 (1933) 255-260. · Zbl 0006.41502 |
| [16] | Hassler Whitney, Tangents to an analytic variety, Ann. of Math. (2) 81 (1965), 496 – 549. · Zbl 0152.27701 · doi:10.2307/1970400 |
| [17] | O. Zariski, “On the problem of existence of algebraic functions of two variables possessing a given branch curve”, Amer. J. Math. 51 (1929) 305-328. · JFM 55.0806.01 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
