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Semicontinuity of the spectrum at infinity. (English) Zbl 0973.32014

Regular functions \(f: U\to {\mathbb C}\) on an affine manifold \(U\), with only isolated critical points are considered. The authors call such a function weakly tame if it is either \(M\)-tame (a notion introduced in A. Némethi and A. Zaharia [Publ. Res. Inst. Math. Sci. 26, No. 4, 681-689 (1990; Zbl 0736.32024)] and analogous to the existence of the Milnor-fibration of hypersurface singularities), or cohomologically tame (a notion which emphasizes the behavior of the function at infinity for some compactification). It is not clear whether one of these properties is stronger than the other.
For \(\gamma\in {\mathbb Q}\), the authors define \(\Sigma_\gamma (f)=\Sigma_{\beta\in(\gamma,\gamma+1]}\nu_\beta(f)\), where \(\beta\mapsto \nu_\beta(f)\) is the integral valued function associated with the spectrum of \(f\) at infinity. (This function was defined in C. Sabbah [C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 7, 603-608 (1999; Zbl 0967.32028)] for cohomologically tame functions and shown to be well-defined for M-tame functions in the present paper.)
The main result of the paper is the following: Let \((S,0)\) be an analytic germ of a smooth curve and let \(\pi: {\mathcal U}\to S\) be a smooth affine morphism with smooth affine fibers \({\mathcal U}_s\) and let \(f: {\mathcal U}\to {\mathbb C}\) be an analytic family of regular functions on \({\mathcal U}_s\). Assume that for every \(s\in S\), the function \(f_s: {\mathcal U}_s\to {\mathbb C}\) is weakly tame. Then for any \(\gamma\in {\mathbb Q}\), \(\Sigma_\gamma (f_0)\leq \Sigma_\gamma(f_s)\).
The authors consider this result as an analogue of the theorem on semi-continuity of the spectrum of an isolated hypersurface singularity proved by A. N. Varchenko [Sov. Math., Dokl. 27, 735-739 (1983; Zbl 0537.14003)] and J. H. M. Steenbrink [Invent. Math. 79, 557-565 (1985; Zbl 0568.14021)].

MSC:

32S05 Local complex singularities
32S25 Complex surface and hypersurface singularities
32S55 Milnor fibration; relations with knot theory
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