Duke Mathematical Journal

Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink

Gil Guibert, François Loeser, and Michel Merle

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Abstract

We prove a motivic analogue of Steenbrink's conjecture [25, Conjecture 2.2] on the Hodge spectrum (proved by M. Saito in [21]). To achieve this, we construct and compute motivic iterated vanishing cycles associated with two functions. We are also led to introduce a more general version of the convolution operator appearing in the motivic Thom-Sebastiani formula. Throughout the article we use the framework of relative equivariant Grothendieck rings of varieties endowed with an algebraic torus action

Article information

Source
Duke Math. J., Volume 132, Number 3 (2006), 409-457.

Dates
First available in Project Euclid: 1 April 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1143935996

Digital Object Identifier
doi:10.1215/S0012-7094-06-13232-5

Mathematical Reviews number (MathSciNet)
MR2219263

Zentralblatt MATH identifier
1173.14301

Subjects
Primary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14B07: Deformations of singularities [See also 14D15, 32S30] 14J17: Singularities [See also 14B05, 14E15] 32S05: Local singularities [See also 14J17] 32S25: Surface and hypersurface singularities [See also 14J17] 32S30: Deformations of singularities; vanishing cycles [See also 14B07] 32S35: Mixed Hodge theory of singular varieties [See also 14C30, 14D07] 32S55: Milnor fibration; relations with knot theory [See also 57M25, 57Q45]

Citation

Guibert, Gil; Loeser, François; Merle, Michel. Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink. Duke Math. J. 132 (2006), no. 3, 409--457. doi:10.1215/S0012-7094-06-13232-5. https://projecteuclid.org/euclid.dmj/1143935996


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References

  • E. Bierstone and P. D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), 207--302.
  • F. Bittner, The universal Euler characteristic for varieties of characteristic zero, Compos. Math. 140 (2004), 1011--1032.
  • —, On motivic zeta functions and the motivic nearby fiber, Math. Z. 249 (2005), 63--83.
  • J. Denef, On the degree of Igusa's local zeta function, Amer. J. Math. 109 (1987), 991--1008.
  • J. Denef and F. Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), 505--537.
  • —, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201--232.
  • —, Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J. 99 (1999), 285--309.
  • —, ``Geometry on arc spaces of algebraic varieties'' in European Congress of Mathematics, Vol. 1 (Barcelona, 2000), Progr. Math. 201, Birkhaüser, Basel, 2001, 327--348.
  • —, Lefschetz numbers of iterates of the monodromy and truncated arcs, Topology 41 (2002), 1031--1040.
  • S. Encinas and H. Hauser, Strong resolution of singularities in characteristic zero, Comment. Math. Helv. 77 (2002), 821--845.
  • S. Encinas and O. Villamayor, Good points and constructive resolution of singularities, Acta Math. 181 (1998), 109--158.
  • W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1984.
  • G. Guibert, Espaces d'arcs et invariants d'Alexander, Comment. Math. Helv. 77 (2002), 783--820.
  • I. N. Iomdin [Yomdin], Complex surfaces with a one-dimensional set of singularities (in Russian), Sibirsk. Mat. Ž. 15 (1974), 1061--1082., 1181; English translation in Siberian Math. J. 15 (1974), 748--762.
  • G. Laumon and L. Moret-Bailly, Champs algébriques, Ergeb. Math. Grenzgeb. (3) 39, Springer, Berlin, 2000.
  • E. Looijenga, Motivic measures, Astérisque 276 (2002), 267--297., Séminaire Bourbaki 1999/2000, no. 874.
  • A. NéMethi and J. H. M. Steenbrink, Spectral pairs, mixed Hodge modules, and series of plane curve singularities, New York J. Math. 1 (1994/95), 149--177.
  • M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), 849--995.
  • —, Duality for vanishing cycle functors, Publ. Res. Inst. Math. Sci. 25 (1989), 889--921.
  • —, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), 221--333.
  • —, On Steenbrink's conjecture, Math. Ann. 289 (1991), 703--716.
  • T. Shioda and T. Katsura, On Fermat varieties, Tôhoku Math. J. (2) 31 (1979), 97--115.
  • D. Siersma, The monodromy of a series of hypersurface singularities, Comment. Math. Helv. 65 (1990), 181--197.
  • J. H. M. Steenbrink, ``Mixed Hodge structure on the vanishing cohomology'' in Real and Complex Singularities (Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, Netherlands, 1977, 525--563.
  • —, ``The spectrum of hypersurface singularities'' in Actes du Colloque de Théorie de Hodge (Luminy, France, 1987), Astérisque 179 --.180, Soc. Math. France, Montrouge, 1989, 11, 163--184.
  • H. Sumihiro, Equivariant completion, II, J. Math. Kyoto Univ. 15 (1975), 573--605.
  • A. N. Varchenko [VarčEnko], Asymptotic Hodge structure on vanishing cohomology (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), 540--591., 688; English translation in Math. USSR-Izv. 18 (1982), 469--512.
  • O. E. Villamayor [Villamayor U.], Constructiveness of Hironaka's resolution, Ann. Sci. École Norm. Sup. (4) 22 (1989), 1--32.
  • —, Patching local uniformizations, Ann. Sci. École Norm. Sup. (4) 25 (1992), 629--677.