La monodromie Hamiltonienne des cycles évanescents

We study the monodromy of vanishing cycles for map-germs \(f:(C^{2n},0) \to (\CM^k,0)\) whose components are in involution. Although the singular fibres of such maps have non-isolated singularities, it is shown that the regular fibres are \(2(n-k)\)-connected and that the vanishing homology group of rank \(2(n-k)+1\) is freely generated by the vanishing cycles. As corollaries, we get that the multiplicity of the discriminant is equal to the dimension of the vanishing homology group of rank \(2(n-k)+1\) and that the Variation operator is an isomorphism. These results are proved under two assumptions: 1. the pyramidality assumptions which states that the singular locus is propagated along the Hamilton flow of the components of \(f\) 2. the generic singular fibres should have transverse Morse singularities and their locus should be connected. It is conjectured that outside a set of infinite codimension the first condition holds and that there exists an involutive deformation of \(f\) which satisfies condition 2.