Singular cohomology of the analytic Milnor fiber, and mixed Hodge structure on the nearby cohomology
Author:
Johannes Nicaise
Journal:
J. Algebraic Geom. 20 (2011), 199-237
DOI:
https://doi.org/10.1090/S1056-3911-10-00526-6
Published electronically:
January 21, 2010
MathSciNet review:
2762990
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Abstract |
References |
Additional Information
Abstract: We describe the homotopy type of the analytic Milnor fiber in terms of a strictly semi-stable model, and we show that its singular cohomology coincides with the weight zero part of the mixed Hodge structure on the nearby cohomology. We give a similar expression for Denef and Loeser’s motivic Milnor fiber in terms of a strictly semi-stable model.
References
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- Michael Temkin, Desingularization of quasi-excellent schemes in characteristic zero, Adv. Math. 219 (2008), no. 2, 488–522. MR 2435647, DOI https://doi.org/10.1016/j.aim.2008.05.006
- Amaury Thuillier, Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels, Manuscripta Math. 123 (2007), no. 4, 381–451 (French, with English summary). MR 2320738, DOI https://doi.org/10.1007/s00229-007-0094-2
References
- J. Ayoub. Les motifs des variétés rigides analytiques. preprint, 2008.
- J. Ayoub. Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. volumes 314-315 of Astérisque, SMF, 2007.
- V. G. Berkovich. A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures. In Algebra, Arithmetic and Geometry - Manin Festschrift (to appear). Boston: Birkhäuser.
- V. G. Berkovich. Spectral theory and analytic geometry over non-archimedean fields, volume 33 of Mathematical Surveys and Monographs. AMS, 1990. MR 1070709 (91k:32038)
- V. G. Berkovich. Étale cohomology for non-Archimedean analytic spaces. Publ. Math., Inst. Hautes Étud. Sci., 78:5–171, 1993. MR 1259429 (95c:14017)
- V. G. Berkovich. Vanishing cycles for formal schemes II. Invent. Math., 125(2):367–390, 1996. MR 1395723 (98k:14031)
- V. G. Berkovich. Smooth $p$-adic analytic spaces are locally contractible. Invent. Math., 137(1):1–84, 1999. MR 1702143 (2000i:14028)
- V. G. Berkovich. An analog of Tate’s conjecture over local and finitely generated fields. Int. Math. Res. Not., 2000(13):665–680, 2000. MR 1772523 (2001h:14022)
- P. Berthelot. Cohomologie rigide et cohomologie rigide à supports propres. Prepublication, Inst. Math. de Rennes, 1996.
- E. Brieskorn. Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscr. Math., 2:103–161, 1970. MR 0267607 (42:2509)
- B. Conrad. Irreducible components of rigid spaces. Ann. Inst. Fourier, 49(2):473–541, 1999. MR 1697371 (2001c:14045)
- A. J. de Jong. Crystalline Dieudonné module theory via formal and rigid geometry. Publ. Math., Inst. Hautes Étud. Sci., 82:5–96, 1995.
- J. Denef and F. Loeser. Motivic Igusa zeta functions. J. Algebraic Geom., 7:505–537, 1998. MR 1618144 (99j:14021)
- J. Denef and F. Loeser. Geometry on arc spaces of algebraic varieties. Progr. Math., 201:327–348, 2001. MR 1905328 (2004c:14037)
- P. Gabriel and M. Zisman. Calculus of fractions and homotopy theory, volume 35 of Ergebnisse der Math. und ihrer Grenzgebiete. Springer-Verlag, 1967. MR 0210125 (35:1019)
- G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat. Toroidal embeddings 1, volume 339 of Lecture Notes in Mathematics. Springer-Verlag, 1973. MR 0335518 (49:299)
- V. Navarro Aznar. Sur la théorie de Hodge-Deligne. Invent. Math., 90:11–76, 1987. MR 906579 (88j:32037)
- V. Navarro Aznar. Sur les structures de Hodge mixtes associées aux cycles evanescents. In Hodge theory, Proc. U.S.-Spain Workshop, Sant Cugat/Spain 1985, volume 1246 of Lect. Notes Math., pages 143–153, 1987. MR 894049 (88i:32036)
- J. Nicaise. A trace formula for rigid varieties, and motivic Weil generating series for formal schemes. Math. Ann., 343:2, 285–349, 2009.
- J. Nicaise and J. Sebag. Invariant de Serre et fibre de Milnor analytique. C.R.Ac.Sci., 341(1):21–24, 2005. MR 2153386 (2006b:14014)
- J. Nicaise and J. Sebag. The motivic Serre invariant, ramification, and the analytic Milnor fiber. Invent. Math., 168(1):133–173, 2007. MR 2285749
- J. Nicaise and J. Sebag. Rigid geometry and the monodromy conjecture. In D. Chéniot et al., editor, Singularity Theory, Proceedings of the 2005 Marseille Singularity School and Conference, pages 819–836. World Scientific, 2007. MR 2342941 (2008h:14021)
- C. Peters and J.H.M. Steenbrink. Mixed Hodge structures, volume 52 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Berlin: Springer, 2008. MR 2393625
- J.H.M. Steenbrink. Limits of Hodge structures. Invent. Math., 31:229–257, 1976. MR 0429885 (55:2894)
- J.H.M. Steenbrink. Mixed Hodge structure on the vanishing cohomology. In P. Holm, editor, Real and complex Singularities, Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, pages 525–563. Alphen a.d. Rijn: Sijthoff & Noordhoff, 1977. MR 0485870 (58:5670)
- M. Temkin. Desingularization of quasi-excellent schemes in characteristic zero. Adv. Math., 219:2, 488–522, 2008. MR 2435647
- A. Thuillier. Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels. Manuscr. Math., 123(4):381–451, 2007. MR 2320738 (2008g:14038)
Additional Information
Johannes Nicaise
Affiliation:
Université Lille 1, Laboratoire Painlevé, CNRS - UMR 8524, Cité Scientifique, 59655 Villeneuve d’Ascq Cédex, France
Address at time of publication:
KULeuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Heverlee, Belgium
MR Author ID:
725638
Email:
johannes.nicaise@math.univ-lille1.fr, johannes.nicaise@wis.kuleuven.be
Received by editor(s):
September 10, 2008
Received by editor(s) in revised form:
January 21, 2009
Published electronically:
January 21, 2010
Additional Notes:
The research for this article was partially supported by ANR-06-BLAN-0183 and ANR-07-JCJC-0004