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On the topology of algebraic varieties. (English) Zbl 0703.14012

Algebraic geometry, Proc. Summer Res. Inst., Brunwick/Maine 1985, part 1, Proc. Symp. Pure Math. 46, No. 1, 15-46 (1987).
[For the entire collection see Zbl 0626.00011.]
These lectures have as aim: to discuss some recent work related to the homotopy groups \(\pi_ i\) of algebraic varieties, with particular attention to \(\pi_ 0\) (connectivity) and \(\pi_ 1\) (fundamental groups):
I. Introduction. An attempt to unify classical theorems of Bézout, Bertini, Lefschetz, Zariski, and Barth leads to a general connectedness principle, which provides the theme of these lectures.
II. Morse theory on analytic spaces: A bit of the history, and an introduction to the work of M. Goresky and R.MacPherson [“Stratified Morse theory” (1988; Zbl 0639.14012)].
III. Lefschetz and connectedness theorems: From this Morse theory one proves a version of Lefschetz’ theorem, from which a general connectedness theorem follows.
IV. \(\pi_ 0\) and varieties of small codimension: Some remarkable applications by Zak and others to problems in classical projective geometry.
V. \(\pi_ 1\) and branched coverings: Applications to fundamental groups and branched coverings of projective space, with an introduction to the work of M. V. Nori [Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 305-344 (1983; Zbl 0527.14016)].
Some of this work has been discussed in other survey lectures. One purpose of these talks is to bring the survey by the author and R. Lazarsfeld in Algebraic geometry, Proc. Conf., Chicago Circle 1980, Lect. Notes Math. 862, 26-91 (1981; Zbl 0484.14005) up to date. The bibliography includes many papers and recent preprints with related and overlapping results.

MSC:

14F45 Topological properties in algebraic geometry
14F35 Homotopy theory and fundamental groups in algebraic geometry
14E20 Coverings in algebraic geometry
14M07 Low codimension problems in algebraic geometry
14H30 Coverings of curves, fundamental group