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Affine Picard-Lefschetz formula. (La formule de Picard-Lefschetz affine.) (French) Zbl 0869.32017
Let \(P:\mathbb{C}^2 \to\mathbb{C}\) be a polynomial of two variables. It defines a locally free fibration outside the critical values of \(P\) and some other points which correspond to singularities at infinity. Let \(\nu_\infty (t_0)\) be the jump of Milnor number at infinity occurring at \(t_0 \in\mathbb{C}\). Huy Vui Ha calls \(t_0\) a critical value of Morse type if \(\nu_\infty (t_0)\) equals 1 or 2. Vanishing cycles and monodromy are studied with the help of a general linear form. This allows also to define a ramification index at \(t_0\). The main result states that the monodromy around a critical value of Morse type is nontrivial if and only if the ramification index is odd and the number of irreducible components of \(P^{-1} (t_0)\) equals that of generic fibres. In this case one has a Picard-Lefschetz formula. For \(\nu_\infty (t_0) =1\) the behaviour can be described with half-cycles analogous to the case of boundary singularities treated by V. I. Arnol’d [Russ. Math. Surv. 33, 99-116 (1978); translation from Usp. Mat. Nauk 33, No. 56(203), 91-105 (1978; Zbl 0408.58009)].

32S30 Deformations of complex singularities; vanishing cycles
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)