Sur certaines singularités d’hypersurfaces 𝐼𝐼

Abstract

The aim of the present article is to construct analytic invariants for a germ of a holomorphic function having a one-dimensional critical locus S S . This is done for a large class of such germs containing for instance any quasi-homogeneous germ at the origin. More precisely, aside from the Brieskorn ( a , b ) (a,b) -module at the origin and a (locally constant along S ∗ := S ∖ { 0 } S^* : = S \setminus \{0\} ) sheaf H n \mathcal {H}^n of ( a , b ) (a,b) -modules associated with the transversal hypersurface singularities along each connected component of S ∗ S^* , we construct also ( a , b ) (a,b) -modules “with supports” E c E_c and E c ∩ S ′ E’_{c \cap \, S} . An interesting consequence of the local study along S ∗ S^* is the corollary showing that for a germ with an isolated singularity, the largest sub- ( a , b ) (a,b) -module having a simple pole in its Brieskorn- ( a , b ) (a,b) -module is independent of the choice of a reduced equation for the corresponding hypersurface germ. We also give precise relations between these various ( a , b ) (a,b) -modules via an exact commutative diagram. This is an ( a , b ) (a,b) -linear version of the tangling phenomenon for consecutive strata we have previously studied in the “topological” setting for the localized Gauss-Manin system of f f . Finally we show that in our situation there exists a non-degenerate ( a , b ) (a,b) -sesquilinear pairing \[ h : E × E c ∩ S ′ ⟶ | Ξ ′ | 2 h : E \times E’_{c\,\cap \, S} \longrightarrow \vert \Xi ’ \vert ^2 \] where | Ξ ′ | 2 \vert \Xi ’ \vert ^2 is the space of formal asymptotic expansions at the origin for fiber integrals. This generalizes the canonical hermitian form defined in 1985 for the isolated singularity case (for the ( a , b ) (a,b) -module version see the recent 2005 paper). Its topological analogue (for the eigenvalue 1 1 of the monodromy) is the non-degenerate sesquilinear pairing \[ h : H c ∩ S n ( F , C ) = 1 × H n ( F , C ) = 1 → C h : H^n_{c\,\cap \,S}(F, \mathbb {C})_{=1} \times H^n(F, \mathbb {C})_{=1} \to \mathbb {C} \] defined in an earlier paper for an arbitrary germ with a one-dimensional critical locus. Then we show this sesquilinear pairing is related to the non-degenerate sesquilinear pairing introduced on the sheaf H n \mathcal {H}^n via the canonical Hermitian form of the transversal hypersurface singularities.