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Semicontinuity of the singularity spectrum. (English) Zbl 0568.14021
To each isolated hypersurface singularity f: (\({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) with Milnor number \(\mu\) one can associate a sequence of \(\mu\) rational numbers: its spectrum. It arises by choosing rationals \(\lambda\) such that exp \(2\pi\) \(i\lambda\) is an eigenvalue of the monodromy operator T, in a way which is determined by the Hodge filtration on the vanishing cohomology of f. In 1980, V. I. Arnol’d [cf. Geometry and analysis, Pap. dedic. Mem. V. K. Patodi, 1-9 (1981; Zbl 0492.58006)] conjectured that the spectrum behaves semicontinuously under deformations of the singularity in a certain way. For a deformation \((f_ t)\) of negative weight of a quasihomogeneous function A. N. Varchenko [Sov. Math., Dohl. 27, 735-739 (1983); translation from Dokl. Akad. Nauk SSSR 270, 1294-1297 (1983; Zbl 0537.14003)] proved the following: let \(x_ 1,...,x_ r\) be singular points of \(f_ t^{- 1}(s)\) (t\(\neq 0)\). Then for any \(a\in {\mathbb{R}}\), the number of spectrum numbers of \(f_ 0\) which lie in \((a,a+1)\) is not less than the analogous number for \(f_ t\), summed over all \(x_ i\). In the paper under review, this result is extended to arbitrary deformations of arbitrary isolated singularities, with \((a,a+1)\) replaced by the half-open interval \((a,a+1]\), and Arnol’d’s original question is answered affirmatively. The proof uses the existence of limit mixed Hodge structures for geometric variations of mixed Hodge structure. A byproduct is the semicontinuity of Hodge numbers for isolated complex intersection singularities and the semicontinuity of the complex singularity index, conjectured by Malgrange.

14J17 Singularities of surfaces or higher-dimensional varieties
14B05 Singularities in algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
Full Text: DOI EuDML
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