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Forme hermitienne canonique sur la cohomologie de la fibre de Milnor d’une hypersurface à singularité isolée. (French) Zbl 0574.32011

The purpose of this article is to show that asymptotic expansions in \(s=0\) of integrals \(\int_{f=s}\phi \quad\) studied by D. Barlet in Invent. Math. 68, 129-174 (1982; Zbl 0508.32003), allows to build up, for an isolated singularity of hypersurface in \({\mathbb{C}}^{n+1}\), a canonical hermitian form on the cohomology of the Milnor fiber of f. This hermitian form is non degenerated (theorem 4) and coincides with the intersection form (in its hermitian version) on characteristic subspaces of the monodromy corresponding to eigenvalues not equal to 1 (theorem 3). This hermitian form is in fact the bridge between the ”classical” theory of the Gauss-Manin system, that is to say the study of integrals of holomorphic forms on horizontal families of cycles, and the asymptotic expansion module at \(s=0\) for integrals \(\int_{f=s}\phi \quad\) where \(\phi\) is \(C^{\infty}\) of type (n,n) with compact support in \({\mathbb{C}}^{n+1}\). For instance, the sesquilinear hermitian quasi- horizontal form on the relative de Rham cohomology which corresponds (via the dictionary of the theorem 2 bis) to the canonical hermitian form, gives a generator system for the asymptotic expansions module (over the power series ring \({\mathbb{C}}[[s,\bar s]])\) in term of a generator system of the relative de Rham cohomoloy (over the ring \({\mathbb{C}}[[s]])\). This reduces, in concrete examples, the computation of all possible asymptotic expansions to the computation of a finite number of expansions for explicit integrals.

MSC:

32S05 Local complex singularities
32C30 Integration on analytic sets and spaces, currents
14J17 Singularities of surfaces or higher-dimensional varieties
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F40 de Rham cohomology and algebraic geometry
32B15 Analytic subsets of affine space

Citations:

Zbl 0508.32003
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Full Text: DOI EuDML

References:

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