La monodromie Hamiltonienne des cycles \'evanescents

Abstract

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.