La monodromie Hamiltonienne des cycles évanescents

2005/05/11 by Mauricio D. Garay, Garay, Mauricio D.
Mathematics · #32S50 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:32S50

paper · pdf · doi:10.48550/arxiv.math/0505209

13 pages

arxiv created 2005/05/11 · arxiv updated 2009/12/01

Abstract

We study the monodromy of vanishing cycles for map-germs f:(C²ⁿ,0) → (\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.

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