Survey on Some Aspects of Lefschetz Theorems in Algebraic Geometry

  • Published 2016


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