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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.

##### MSC:
 14J17 Singularities of surfaces or higher-dimensional varieties 14B05 Singularities in algebraic geometry 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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##### References:
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