## 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)

### Keywords:

spectrum of hypersurface singularities; Milnor number

### Citations:

Zbl 0695.00012; Zbl 0325.32003