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On certain hypersurface singularities. II. (Sur certaines singularités d’hypersurfaces. II.) (French) Zbl 1138.32015

Summary: The aim of the present article is to construct analytic invariants for a germ of a holomorphic function having a one-dimensional critical locus \(S\). 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 \((a,b)\)-module at the origin and a (locally constant along \(S^* := S\{0\}\)) sheaf \({\mathcal H}^n\) of \((a, b)\)-modules associated with the transversal hypersurface singularities along each connected component of \(S^*\), we construct also \((a,b)\)-modules with supports \(E_c\) and \(E'_{c\cap S}\).
An interesting consequence of the local study along \(S^*\) is the corollary showing that for a germ with an isolated singularity, the largest sub-\((a, b)\)-module having a simple pole in its Brieskorn-\((a,b)\)-module is independent of the choice of a reduced equation for the corresponding hypersurface germ.
We also give precise relations between these various \((a,b)\)-modules via an exact commutative diagram. This is an \((a, b)\)-linear version of the tangling phenomenon for consecutive strata we have previously studied in the “topological” setting for the localized Gauss-Manin system of \(f\).
Finally, we show that in our situation there exists a non-degenerate \((a, b)\)-sesquilinear pairing
\[ h:E\times E_{e\cap S}'\to |\Xi'|^2 \]
where \(||\Xi'|^2\) 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 \((a, b)\)-module version see the recent 2005 paper). Its topological analogue (for the eigenvalue 1 of the monodromy) is the non-degenerate sesquilinear pairing
\[ h: H^n_{c\cap S}(F,\mathbb C)_{=1}\times H^m(F,\mathbb C)_{=1}\to \mathbb C \]
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 \({\mathcal H}^n\) via the canonical Hermitian form of the transversal hypersurface singularities. For part I of this paper see [D. Bartlet, Bull. Soc. Math. Fr. 134, No. 2, 173–200 (2006; Zbl 1126.32022).]

MSC:

32S25 Complex surface and hypersurface singularities
32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
14B05 Singularities in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties

Citations:

Zbl 1126.32022
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References:

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