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The spectrum of hypersurface singularities. (English) Zbl 0725.14031

Théorie de Hodge, Actes Colloq., Luminy/Fr. 1987, Astérisque 179-180, 163-184 (1989).
[For the entire collection see Zbl 0695.00012.]
The author defines the spectrum for arbitrary (not necessarily isolated) hypersurface singularities and investigates some of its properties. Let \({\mathcal S}={\mathbb{Z}}^{({\mathbb{Q}})}\) be the free abelian group on generators (\(\alpha\)), \(\alpha\in {\mathbb{Q}}\). A typical element of \({\mathcal S}\) is denoted by \(\sum n_{\alpha}(\alpha) \). Let \({\mathcal C}\) denote the category whose objects are \({\mathbb{C}}[t]\)-modules of finite length equipped with t-stable decreasing filtrations on which t acts as an automorphism of finite order and the morphisms of which are \({\mathbb{C}}[t]\)-linear maps which are compatible with the given filtrations. A typical object of \({\mathcal C}\) is denoted as (H,F,\(\gamma\)) where F is the filtration and \(\gamma\) the automorphism given by the action of t. Put \(s(*p)=\dim (Gr^ p_ F(H))\) and define rational numbers \(\alpha_ 1,...,\alpha_{s(p)}\) by \(n-p-1<\alpha_ j\leq n-p\); \[ \det (tI- \gamma;\;Gr^ p_ F(H))=\prod^{s_ p}_{j=1}(t-e^{-2i\pi \alpha_ j}). \] Put \(Sp_ n(H,F,\gamma):=\sum_{p}\sum^{s_ p}_{j=1}(\alpha_ j).\) Thus \(Sp_ n(H,F,\gamma)\) is an element in \({\mathcal S}.\)
Let f: (\({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) be a non-zero holomorphic function germ. Let X(f) be its Milnor fibre. The cohomology groups \(H^*(X(f))\) carry a canonical mixed Hodge structure and the semi-simple part \(T_ s\) of the monodromy acts as an automorphism of these mixed Hodge structures. In particular, it preserves the Hodge filtration F. The spectrum of f is defined by \(Sp(f):=\sum^{n}_{k=0}(-1)^{n-k}Sp_ n(\tilde H^ k(X(f)),F,T_ s) \). - For example, for \(f=z_ 0^{k+1}+z^ 2_ 1...+z^ 2_ n\) the spectrum is \(\sum^{k}_{i=1}((k+1)^{-1}+n/2-1).\)
In the present paper the author considers a hypersurface \(\{f=0\}\) in \({\mathbb{C}}^{n+1}\) the singular locus of which is of dimension one and compares this with a hypersurface \(\{f+\epsilon \ell^ k=0\}\) where \(\epsilon\) is sufficiently small and \(\ell\) is a linear form which is not tangent to any component of the critical locus of \(f.\) The author gives a conjecture on the spectrum of \(f+\epsilon \ell^ k\) (conjecture (2.2)) which generalizes a formula of I. N. Iomdin [Sib. Math. J. 15 (1974), 748-762 (1975); translation from Sib. Mat. Zh. 15, 1061-1082 (1974; Zbl 0325.32003)] on the Milnor number. In the present paper the author proves the conjecture for certain cases.
Recently M. Saito [“Vanishing cycles and mixed Hodge modules”, Preprint Inst. Hautes Étud. Sci. M/88/41 (1988)] proved the conjecture.
Reviewer: K.Ueno (Kyoto)

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
14B05 Singularities in algebraic geometry
32S25 Complex surface and hypersurface singularities
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)