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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)].

##### MSC:
 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.)