×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [A] Arnol’d, V.I.: On some problems in singularity theory. Geometry and Analysis, Papers dedicated to the memory of V.K. Patodi, Bombay 1981, pp. 1-10
[2] [B] du Bois, Ph.: Structure de Hodge mixte sur la cohomologie évanescente. Ann. Inst. Fourier (to appear) · Zbl 0535.14004
[3] [D] Deligne, P.: Théorie de Hodge. II. Publ. Math. IHES40, 5-58 (1971); III Publ. Math. IHES44, 5-75 (1975) · Zbl 0219.14007
[4] [Du] Durfee, A.: The signature of smoothings, of complex surface singularities. Math. Ann.232, 85-98 (1978) · Zbl 0357.32008
[5] [E] Elkik, R.: Singularités rationelles et déformations. Invent. Math.47, 139-147 (1978) · Zbl 0383.14005
[6] [EZ] El Zein, F.: Dégénérescence diagonale. I. C.R. Acad. Sc. Paris276, 51-54 (1983); II. C.R. Acad. Sc. Paris276, 199-202 (1983) · Zbl 0538.14004
[7] [G] Greuel, G.-M.: Der Gauss-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten. Math. Ann.214, 235-266 (1975) · Zbl 0292.14007
[8] [GNP] Guillén, F., Navarro Aznar, V., Puerta, F.: Théorie de Hodge via schémas cubiques. Mimeographed notes, Barcelona 1982
[9] [GS] Greuel, G.-M., Steenbrink, J.H.M.: On the topology of smoothable singularities. Proc. Symp. Pure Math.40, Part I, 535-545 (1983)
[10] [M] Milnor, J.: Singular points of complex hypersurfaces. Annals of Math. Studies vol. 61, Princeton 1968 · Zbl 0184.48405
[11] [Ma] Malgrange, B.: Intégrales asymptotiques et monodromie. Ann. Sc. Éc. Norm. Sup. 4e série,7, 405-430 (1974)
[12] [Sc] Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math.22, 211-320 (1973) · Zbl 0278.14003
[13] [S1] Steenbrink, J.H.M.: Mixed Hodge structure on the vanishing cohomology. Real and complex singularities, Oslo 1976. Sijthoff-Noordhoff 1977, pp. 525-563
[14] [S2] Steenbrink, J.H.M.: Mixed Hodge structures associated with isolated singularities. Proc. Symp. Pure Math.40, Part II, 513-536 (1983)
[15] [SZ] Steenbrink, J.H.M., Zucker, S.: Variation of mixed Hodge structure I. Univ. of Leiden, Report 2, Jan. 1984 · Zbl 0626.14007
[16] [V1] Varchenko, A.N.: The complex exponent of a singularity does not change along strata ? =const. Funct. An. Appl.16, 1-10 (1982) · Zbl 0498.32010
[17] [V2] Varchenko, A.N.: On semicontinuity of the spectrum and an upper bound for the number of singular points of projective hypersurfaces. Dokladyi Ak. Nauk.270 (6) 1294-1297 (1983)
[18] [V3] Varchenko, A.N.: Asymptotics of integrals and Hodge structures. In: Modern Problems of Mathematics, vol.22, pp. 130-166 (1983) (in Russian)
[19] [V4] Varchenko, A.N.: On change of discrete invariants of critical points of functions under deformation. Uspehi Mat. Nauk.5, 126-127 (1983)
[20] [V5] Varchenko, A.N.: On semicontinuity of the complex singularity index. Funct. An. Appl.17, 77-78 (1983)
[21] [V6] Varchenko, A.N.: Asymptotic Hodge structure in the vanishing cohomology. Math. USSR Izvestija18, 469-512 (1982) · Zbl 0489.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.