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Mixed Hodge Structures Applied to Singularities

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Handbook of Geometry and Topology of Singularities III

Abstract

We give a survey of applications of mixed Hodge theory to the study of isolated singularities. A summary of mixed Hodge theory is followed by some examples. The formalism of vanishing cycles is realized on the Hodge level by a sheaf complex with three filtrations, making the application to the cohomology of the Milnor fibre possible. The approaches by algebraic analysis and by motivic integration are discussed, and the spectrum with its properties is considered. The paper ends with a treatment of Du Bois singularities.

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References

  1. V. I. Arnol’d. Index of a singular point of a vector field, the Petrovskii-Oleinik inequality, and mixed Hodge structures. Funct. Anal. Appl. 12, 1–12, 1978.

    Article  Google Scholar 

  2. V. I. Arnol’d. On some problems in singularity theory. Proc. Indian Acad. Sci. (Math. Sci.), Vol. 90, number 1, 1–9, 1981.

    Article  MathSciNet  Google Scholar 

  3. S. Baljonan, C. Hertling. Real Seifert Forms and Polarizing Forms of Steenbrink Mixed Hodge Structures. Bull. Braz. Math. Soc., New Series 50, 233–274, 2019.

    Google Scholar 

  4. F. Bittner. The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140, 1011–1032, 2004.

    Article  MathSciNet  Google Scholar 

  5. F. Bittner. On motivic zeta function and the motivic nearby fibre. Math. Z. 249, 63–83, 2005.

    Article  MathSciNet  Google Scholar 

  6. Ph. du Bois. Complexe de de Rham filtré d’une variété singulière. Bull. Soc. Math. France 109, 41–81, 1981.

    Article  MathSciNet  Google Scholar 

  7. Ph. du Bois. Structure de Hodge mixte sur la cohomologie évanescente. Ann. Inst. Fourier, 35(1), 191–213, 1985.

    Article  MathSciNet  Google Scholar 

  8. J.-P. Brasselet, J. Schürmann and S. Yokura. Hirzebruch classes and motivic Chern classes for singular spaces. J. Topol. Anal. 2, no. 1,1–55, 2010.

    Google Scholar 

  9. T. Brélivet. The Hertling conjecture in dimension 2. Preprint. arXiv: math/0405489

    Google Scholar 

  10. T. Brélivet and C. Hertling. Bernoulli moments of spectral numbers and Hodge numbers. Journal of Singularities 20, 205–231, 2020.

    MathSciNet  MATH  Google Scholar 

  11. E. Brieskorn. Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscripta math. 2, 103–161, 1970.

    Article  MathSciNet  Google Scholar 

  12. J.-L. Brylinski. Modules holonomes à singularités régulières et filtration de Hodge. I. In: Algebraic Geometry, La Rabida 1981. Lecture Notes in Math. 961, 1–21. Springer-Verlag Berlin Heidelberg New York 1982.

    Google Scholar 

  13. J. Carlson. Extensions of mixed Hodge structures. In: Géométrie algébrique, Angers 1979. Sijthoff-Noordhoff, Alphen a/d Rijn, 107–127, 1980.

    Google Scholar 

  14. V. I. Danilov. Newton polyhedra and vanishing cohomology. Funct. Anal. Appl. 13, 103–115, 1979.

    Article  MathSciNet  Google Scholar 

  15. P. Deligne. Équations Différentielles à Points Singuliers Réguliers. Lecture Notes in Mathematics, vol. 163. Springer-Verlag Berlin Heidelberg New York 1970.

    Google Scholar 

  16. P. Deligne. Travaux de Griffiths. Sém. Bourbaki, 22e année, 1969/70, no 376.

    Google Scholar 

  17. P. Deligne. Théorie de Hodge II. Publ. Math. IHES 40, 5–58, 1971.

    Article  Google Scholar 

  18. P. Deligne. Théorie de Hodge III. Publ. Math. IHES 44, 5–77, 1974.

    Article  Google Scholar 

  19. J. Denef, F. Loeser. Geometry of arc spaces of algebraic varieties. European Congress of Mathematics, vol. I (Barcelona 2000). Progr. Math. vol. 201, pp. 327–348. Birkhäuser, Basel 2001.

    Google Scholar 

  20. A. Dimca. Monodromy and Hodge theory of regular functions. D. Siersma et al. (eds.), New Developments in Singularity Theory, 257–278, Kluwer 2001.

    Google Scholar 

  21. W. Ebeling and J. H. M. Steenbrink. Spectral pairs for isolated complete intersection singularities. J. Algebraic Geom. 7, 55–76, 1998.

    MathSciNet  MATH  Google Scholar 

  22. Ph. Griffiths. On the periods of certain rational integrals: I. Annals of math. 2nd series Vol. 90 no. 3, 460–495 1969.

    Google Scholar 

  23. Ph. Griffiths. Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems. Bull. Amer. Math. Soc. 76, 228–296, 1970.

    Article  MathSciNet  Google Scholar 

  24. A. Grothendieck. On the de Rham cohomology of algebraic varieties. Publ. Math. IHES 29, 95–103, 1966.

    Article  MathSciNet  Google Scholar 

  25. G. Guibert. Espaces d’arcs et invariants d’Alexander. Comment. Math. Helv. 77, 783–820, 2002.

    Article  MathSciNet  Google Scholar 

  26. G. Guibert, F. Loeser, M. Merle. Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink. Duke Math. J., 132(3), 409–457, 2006.

    Article  MathSciNet  Google Scholar 

  27. F. Guillén, V. Navarro Aznar. Sur le théorème local des cycles invariants. Duke Math. J. 61, 133–155, 1990.

    Article  MathSciNet  Google Scholar 

  28. F. Guillén, V. Navarro Aznar, P. Pascal-Gainza, F. Puerta. Hyperrésolutions cubiques et descente cohomologique. Lecture Notes in Math. 1335, Springer Verlag, Berlin etc. 1988.

    Google Scholar 

  29. C. Hertling. Frobenius manifolds and variance of the spectral numbers. In: D. Siersma et al. (eds.), New Developments in Singularity Theory, 235–255. Kluwer 2001.

    Google Scholar 

  30. F. Hirzebruch, K. H. Mayer. O(n)-Mannigfaltigkeiten, exotische Sphären und Singularitäten. Lecture Notes in Math. 57. Springer-Verlag Berlin Heidelberg New York 1968.

    Google Scholar 

  31. W. V. D. Hodge. The Theory and Applications of Harmonic Integrals. Cambridge Univ. Press, 1947.

    Google Scholar 

  32. M. Kashiwara. On the maximally overdetermined systems of linear differential equations I. Publ. RIMS Kyoto Univ. 10, 563–579, 1975.

    Article  MathSciNet  Google Scholar 

  33. M. Kashiwara and T. Kawai. On holonomic systems of micro-differential equations III. - Systems with regular singularities. Publ. RIMS Kyoto Univ. 17, 813–979 (1981).

    Article  Google Scholar 

  34. J. Kollár and S. Kovács. Log canonical singularities are du Bois. J. Amer. Math. Soc. 23(3), 791–813, 2010.

    Article  MathSciNet  Google Scholar 

  35. S. J. Kovács. Rational, log canonical, Du Bois singularities: on the conjectures of Kollár and Steenbrink. Compos. Math. 118, no. 2, 123–133, 1999.

    Google Scholar 

  36. Va. Kulikov. Mixed Hodge structures and singularities. Cambridge Tracts in Math. 132, Cambridge University Press, Cambridge 1998.

    Google Scholar 

  37. G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat. Toroidal embeddings I. Lecture Notes in Math. 339. Springer, Berlin Heidelberg New York 1973.

    Google Scholar 

  38. B. Malgrange. Intégrales asymptotiques et monodromie. Ann. Scient. Ec. Norm. Sup. 4e série, 7, 405–430, 1974.

    Google Scholar 

  39. L. Maxim, M. Saito and J. Schürmann. Spectral Hirzebruch-Milnor classes of singular hypersurfaces. Math.Ann. 377, no. 1–2, 281–315, 2020.

    Google Scholar 

  40. L. Maxim, M. Saito and J. Schürmann. Thom-Sebastiani theorems for filtered D-modules and for multiplier ideals. Int. Math. Res. Not. IMRN 2020, no. 1, 91–111.

    Google Scholar 

  41. Z. Mebkhout. Une équivalence de catégories. Compos. Math. 51, 51–62. Une autre équivalence de catégories. Ibid. 63–88, 1984.

    Google Scholar 

  42. C. A. M Peters and J. H. M. Steenbrink. Mixed Hodge Structures. Ergebnisse der Mathematik und ihrer Grenzgebiete Volume 52, Springer-Verlag Berlin Heidelberg 2008.

    Google Scholar 

  43. C. A. M Peters and J. H. M. Steenbrink. Hodge number polynomials for nearby and vanishing cohomology. In: J. Nagel, C. Peters (eds.) Algebraic cycles and motives, Volume 2. London Math. Soc. Lect. Note series, vol. 344, 289–303. Cambridge University Press, 2007.

    Google Scholar 

  44. F. Pham. Singularités des systèmes différentiels de Gauss-Manin. Progress in Math. 2, Birkhäuser Verlag, 1979.

    Google Scholar 

  45. M. Saito. On the exponents and the geometric genus of an isolated hypersurface singularity. Proc. Symp. Pure Math. 40, Part 2, 465–472, 1983.

    Google Scholar 

  46. M.Saito. Mixed Hodge modules. Proc. Japan Acad. 62, Ser. A, 360–363, 1986.

    Google Scholar 

  47. M. Saito. Exponents and Newton polyhedra of isolated hypersurface singularities. Math. Ann. 281, 411–417, 1988.

    Article  MathSciNet  Google Scholar 

  48. M. Saito. Mixed Hodge modules. Publ. Res. Inst. Math. Sci. Kyoto Univ. 26, 221–333, 1989.

    Article  MathSciNet  Google Scholar 

  49. M. Saito. On Steenbrink’s conjecture. Math. Ann. 289, 703–716, 1991.

    Article  MathSciNet  Google Scholar 

  50. M. Saito. Exponents of an irreducible plane curve singularity. arXiv.math.AG/0009133.

    Google Scholar 

  51. J. Scherk and J. H. M. Steenbrink. On the mixed Hodge structure on the cohomology of the Milnor fibre. Math. Ann. 271, 641–665, 1985.

    Article  MathSciNet  Google Scholar 

  52. W. Schmid. Variation of Hodge structure: the singularities of the period mapping.. Invent. math. 22, 211–319, 1973.

    Google Scholar 

  53. K. Schwede. A simple characterization of du Bois singularities. Compos. Math. 143, 813–828, 2007.

    Article  MathSciNet  Google Scholar 

  54. D. Siersma, J.H.M Steenbrink and M. Tibar. Polar degree and Huh’s conjectures. J. Algebraic Geom. 30, 189–203, 2021.

    Google Scholar 

  55. J. H. M. Steenbrink. Limits of Hodge structures. Invent. Math, 41, 229–257, 1976.

    Article  MathSciNet  Google Scholar 

  56. J. H. M. Steenbrink. Mixed Hodge structures on the vanishing cohomology. In: P. Holm ed., Real and Complex Singularities, Oslo 1976. Sijthoff-Noordhoff, Alphen a/d Rijn, 525–563, 1977.

    Google Scholar 

  57. J. H. M. Steenbrink. Intersection form for quasi-homogeneous singularities. Compos. Math. 42, 211–223 (1977).

    MathSciNet  MATH  Google Scholar 

  58. J. H. M. Steenbrink. Mixed Hodge structures associated with isolated singularities. Proc. Symp. Pure Math. 40, Part 2, 513–536, 1983.

    Google Scholar 

  59. J. H. M. Steenbrink. Semicontinuity of the singularity spectrum. Invent. math. 79, 557–565, 1985.

    Article  MathSciNet  Google Scholar 

  60. J. H. M. Steenbrink. Vanishing theorems on singular spaces. Astérisque 130, 330–341, 1985.

    MathSciNet  MATH  Google Scholar 

  61. J. H. M. Steenbrink. The spectrum of hypersurface singularities. In: Théorie de Hodge, Luminy, Juin 1987. Astérisque 179–180, 163–184, 1989.

    Google Scholar 

  62. J. H. M. Steenbrink. Monodromy and weight filtration for smoothings of isolated singularities. Compos. Math. 97, 285–293 (1995).

    MathSciNet  MATH  Google Scholar 

  63. J. H. M. Steenbrink. Du Bois invariants of isolated complete intersection singularities. Ann. Inst. Fourier, Grenoble 47(5), 1367–1377, 1997.

    Google Scholar 

  64. J. H. M. Steenbrink. Motivic Milnor fibre for nondegenerate function germs on toric singularities. In: D. Ibadula, W. Veys (eds.), Bridging algebra, geometry, and topology. Springer Proceedings in Mathematics and Statistics 96, Springer 2014.

    Google Scholar 

  65. A. N. Varchenko. Hodge properties of the Gauss-Manin connection. Funct. Anal. Appl. 14, 46–47, 1980.

    Article  MathSciNet  Google Scholar 

  66. A. N. Varchenko. On the monodromy operator in vanishing cohomology and the operator of multiplication by f in the local ring. Sov. Math. Dokl. 24, 248–252, 1981.

    MATH  Google Scholar 

  67. A. N. Varchenko. Asymptotic Hodge structure in vanishing cohomologies. Izv. Akad.Nauk SSSR, Ser. Mat., 45, No. 3, 540–591, 1981.

    Google Scholar 

  68. A. N. Varchenko. The complex exponent of a singularity does not change along strata μ =  const. Funct, Anal, Appl. 16,1, 1–12, 1982.

    Google Scholar 

  69. A. N. Varchenko. On semicontinuty of the spectrum and an upper bound for the number of singular points of projective hypersurfaces. Sov. Math. Dokl. 27, 735–739, 1983.

    MATH  Google Scholar 

  70. A. N. Varchenko. Asymptotic integrals and Hodge structures. J. Soviet Math. 27, 2760–2784, 1984.

    Article  Google Scholar 

  71. A. N. Varchenko and A. G. Khovanskii. Asymptotics of integrals over vanishing cycles and the Newton polyhedron. Soviet Math. Dokl. 32, No. 1, 122–127, 1985.

    Google Scholar 

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Authors and Affiliations

  1. IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands

    Joseph Steenbrink

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  1. Joseph Steenbrink
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Correspondence to Joseph Steenbrink .

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Editors and Affiliations

  1. Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México and, Unidad Mixta Internacional CNRS Laboratorio Solomon Lefschetz, Cuernavaca, Mexico

    José Luis Cisneros-Molina

  2. Centre de Mathématiques et Informatique, Universite d’Aix-Marseille, Marseille, France

    Lê Dũng Tráng

  3. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    José Seade

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Steenbrink, J. (2022). Mixed Hodge Structures Applied to Singularities. In: Cisneros-Molina, J.L., Dũng Tráng, L., Seade, J. (eds) Handbook of Geometry and Topology of Singularities III. Springer, Cham. https://doi.org/10.1007/978-3-030-95760-5_9

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