On the mixed Hodge structure on the cohomology of the Milnor fibre. (English) Zbl 0618.14002

The aim of the paper under review is to introduce a mixed Hodge structure on the cohomology of the Milnor fibre of an isolated hypersurface singularity. Such an attempt was already done earlier in 1976 [see J. H. M. Steenbrink in Real and complex singularities, Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 525-563 (1977; Zbl 0373.14007)] by using completely different methods. The main difference here is that the authors use the theory of holonomic D-modules in one variable with regular singularity (instead of using the resolution of singularities). This second approach has the advantage that one can better understand the Hodge filtration and it is more suited for applications. At the end of the paper the mixed Hodge structure is explicitly computed on two examples.
Reviewer: L.Bădescu


14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
14B05 Singularities in algebraic geometry
32S05 Local complex singularities


Zbl 0373.14007
Full Text: DOI EuDML


[1] Brieskorn, E.: Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscr. Math.2, 103-161 (1970) · Zbl 0186.26101
[2] Brylinski, J.-L.: Modules holonomes à singularités régulieres et filtration de Hodge. I. In: Algebraic Geometry, La Rábida 1981. Lect. Notes Math. 961, 1-21. Berlin, Heidelberg, New York: Springer 1982
[3] Deligne, P.: Equations différentielles à points singuliers réguliers. Lect. Notes Math. 163. Berlin, Heidelberg, New York: Springer 1970 · Zbl 0244.14004
[4] Katz, N.: The regularity theorem in algebraic geometry. Actes du ICM, Vol. I, 437-443, Nice 1970
[5] Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math.32, 1-31 (1976) · Zbl 0328.32007
[6] Malgrange, B.: Intégrales asymptotiques et monodromie. Ann. Sci. Ec. Norm. Super. IV Sér7, 405-430 (1974) · Zbl 0305.32008
[7] Pham, F.: Caustiques, phase stationnaire et microfonctions. Acta Math. Vietn.2, 35-101 (1977) · Zbl 0431.58018
[8] Pham, F.: Singularités des systèmes différentiels de Gauss-Manin. Progress in Math., Vol. 2. Boston: Birkhäuser 1979
[9] Pham, F.: Structures de Hodge mixtes associées à un germe de fonction à point critique isolé. In: Analyse et topologie sur les espaces singuliers (II?III). Asterisque101-102, 268-285 (1983)
[10] Saito, M.: Hodge filtrations on Gauss-Manin systems. I. J. Fac. Sci. Univ. Tokyo, Sect. IA30, 489-498 (1984) · Zbl 0549.32016
[11] Saito, M.: Gauss-Manin system and mixed Hodge structure. Proc. Japan Acad., Ser. A58, 29-32 (1982) · Zbl 0516.32012
[12] Saito, M.: Exponents and Newton polyhedra of isolated hypersurface singularities. Preprint Institut Fourier, Grenoble 1983 · Zbl 0628.32038
[13] Scherk, J.: On the Gauss-Manin connection of an isolated hypersurface singularity. Math. Ann.238, 23-32 (1980) · Zbl 0409.32004
[14] Scherk, J.: On the monodromy theorem for isolated hypersurface singularities. Invent. Math.58, 289-301 (1980) · Zbl 0432.32010
[15] Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math.22, 211-320 (1973) · Zbl 0278.14003
[16] Shioda, T., Katsura, T.: On Fermat varieties. Tôhoku Math. J.31, 97-115 (1979) · Zbl 0415.14022
[17] Steenbrink, J.H.M.: Limits of Hodge structures. Invent. Math.31, 229-257 (1976) · Zbl 0312.14007
[18] Steenbrink, J.H.M.: Mixed Hodge structure on the vanishing cohomology. In: Holm, P., ed. Real and complex singularities, pp. 525-563. Oslo 1976. Alphen aan de Rijn, Sijthoff-Noordhoff 1977
[19] Steenbrink, J.H.M.: Intersection form for quasi-homogeneous singularities. Compos. Math.34, 211-223 (1977) · Zbl 0347.14001
[20] Steenbrink, J.H.M.: Semicontinuity of the singularity spectrum. Invent. Math.79, 557-565 (1985) · Zbl 0568.14021
[21] Varchenko, A.N.: Gauss-Manin connection of isolated singular point and Bernstein polynomial. Bull. Sci. Math., II. Ser.104, 205-223 (1980) · Zbl 0434.32008
[22] Varchenko, A.N.: Hodge properties of the Gauss-Manin connection. Funct. Anal. Appl.14, 46-47 (1980) · Zbl 0457.32008
[23] Varchenko, A.N.: The asymptotics of holomorphic forms determine a mixed Hodge structure. Sov. Math. Dokl.22, 772-775 (1980) · Zbl 0516.14007
[24] Varchenko, A.N.: On the monodromy operator in vanishing cohomology and the operator of multiplication byf in the local ring. Sov. Math. Dokl.24, 248-252 (1981) · Zbl 0497.32007
[25] Varchenko, A.N.: Asymptotic Hodge structure in the vanishing cohomology. Math. USSR Izv.18, 469-512 (1982) · Zbl 0489.14003
[26] Varchenko, A.N.: The complex exponent of a singularity does not change along strata ?=const. Funct. Anal. Appl.16, 1-10 (1982) · Zbl 0498.32010
[27] Varchenko, A.N.: On semicontinuity of the spectrum and an upper bound for the number of singular points of projective hypersurfaces. Dokl. Ak. Nauk.270, 1294-1297 (1983)
[28] Sebastiani, M.: Preuve d’une conjecture de Brieskorn. Manuscr. Math.2, 301-308 (1970) · Zbl 0194.11402
[29] Sebastiani, M., Thom, R.: Un résultat sur la monodromie. Invent. Math.13, 90-96 (1971) · Zbl 0233.32025
[30] Arnol’d, V.I.: On some problems in singularity theory. In: Geometry and analysis. Papers dedicated to the memory of V.K. Patodi, pp. 1-10. Bombay 1981
[31] Goryunov, V.: Adjoining spectra of certain singularities (in Russian). Vestn. Mosc. Univ. Ser. 19814 · Zbl 0514.32004
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.