Survey on Some Aspects of Lefschetz Theorems in Algebraic Geometry

  • HÉLÈNE ESNAULT
  • Published 2016

Abstract

We survey classical material around Lefschetz theorems for fundamental groups, and show the relation to parts of Deligne’s program in Weil II. 1. Classical notions Henri Poincaré (1854-1912) in [Poi95] formalised the notion of fundamental group of a connected topological space X. It had appeared earlier on, notably in the work of Bernhard Riemann (1826-1866) ([Rie51], [Rie57]) in the shape of multi-valued functions. Fixing a base point x ∈ X, then π 1 (X,x) is first the set of homotopy classes of loops centered at x. It has a group structure by composing loops centered at x. It is a topological invariant, i.e. depends only on the homeomorphism type ofX. It is functorial: if f : Y → X is a continuous map, and y ∈ Y , then f induces a homomorphism f∗ : π top 1 (Y, y) → π 1 (X, f(y)) of groups. It determines the topological coverings ofX as follows: fixing x, there is a universal covering Xx, together with a covering map π : Xx → X, and a lift x̃ of x on Xx, such that π top 1 (X,x) is identified with Aut(Xx/X). Let us assumeX is a smooth projective curve over C, that is X(C) is a Riemann surface. By abuse of notations, we write π 1 (X,x) instead of π top 1 (X(C), x). Then π top 1 (X,x) = 0 for P1, the Riemann sphere, that is if the genus g of X is 0, it is equal to Z2 if g = 1 and else for g ≥ 2, it is spanned by 2g generators αi, βi, i = 1, . . . , g with one relation ∏g i=1[αi, βi] = 1. So it is nearly a free group, in fact, for any choice of s points a1, . . . , as of X(C) different from x, s ≥ 1, π 1 (X \{a1, . . . , as}, x) is free spanned by αi, βi, γ1, . . . , γs, with one relation ∏g i=1[αi, βi] ∏s j=1 γj = 1 ([Hat02]). For s = 1, the map π top 1 (X \ a, x) → π top 1 (X,x) is surjective and yields the presentation. More generally, for any non-trivial Zariski open subvariety U →֒ X, the homomorphism π 1 (U, x) → π top 1 (X,x) is always surjective, as we see taking loops and moving them via homotopies inside of U . The kernel in general is more complicated, but is spanned by loops around the divisor at infinity. If X has dimension ≥ 2, then π 1 (X,x) is far from being free. A natural question is how to compute it. This is the content of the Lefschetz (Salomon Lefschetz (1884-1972)) theorems for the fundamental group. Date: March 5, 2016. Supported by the Einstein program.

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Cite this paper

@inproceedings{ESNAULT2016SurveyOS, title={Survey on Some Aspects of Lefschetz Theorems in Algebraic Geometry}, author={H{\'E}L{\`E}NE ESNAULT}, year={2016} }