Sur la topologie des polynômes complexes

Vanishing cycles and monodromy of complex polynomials

“…By Theorem 1.6, this generalises Michel and Weber's positive answer in [11] to Dimca's question of whether the local monodromy around a reduced and irreducible irregular fibre of a polynomial f : C 2 → C must be nontrivial (the same answer is implicit in Theorem 1 of [2]). The conditions are needed here: one can find f : C 2 → C having an irregular fibre with trivial local monodromy, this fibre having any number of components.…”

“…The following theorem localizes the computation of H * (N(c), ∂N(c)) into the singular fibre. Under the assumptions that F has homology only in dimension (n−1) and that all singularities of fibres of f are isolated, Artal-Bartolo, Cassou-Nogués, and Dimca [2] proved the dimension formulae for Ker 1 − h n−1 (c) and H n−1 (F ; Z) that follow from the above theorems. Polynomials f (x 1 , .…”

On the geometry of regular maps from a quasi-projective surface to a curve

“…We consider homology and cohomology with coefficients in C. The monodromy groups Mon k f and Mon k f , k = 0, 1, 2, are by definition the monodromy representations of the fundamental groups π 1 F)), respectively. We are more specifically interested in the first cohomology group and its global invariant cycles, denoted by H 1 (F) π and H 1 (F) π , respectively.…”

“…By using the above cited result of Kaliman and the celebrated theorem of Abhyankar-Moh-Suzuki, Dimca also observed that, in addition to the triviality of the monodromy, if all fibres of the polynomial P are irreducible then P is linearisable. In [1,5,[14][15][16]] the monodromy of a polynomial function on C 2 in relation to the monodromy of a plane curve germ as well as extensions to several variables were studied.…”

Rational polynomials of simple type

“…The classification of rational polynomials of simple type gained some new interest through the result of Cassou-Nogues, Artal-Bartolo, and Dimca [4] that they are precisely the polynomials whose homological monodromy is trivial (it suffices that the homological monodromy at infinity be trivial by an observation of Dimca).…”