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Brieskorn Lattices and Torelli Type Theorems for Cubics in ℙ3 and for Brieskorn-Pham Singularities with Coprime Exponents

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Singularities

Part of the book series: Progress in Mathematics ((PM,volume 162))

Abstract

The first part is a survey about polarized mixed Hodge structures and Brieskorn lattices for hypersurface singularities. It describes wellknown properties of these objects and contains new results about classification spaces and moduli spaces for these objects. In the second part the Brieskorn lattices of cubics in ℂ4 and of Brieskorn-Pham singularities with coprime exponents are studied. A nice application is a global Torelli theorem for cubics in ℙ3 by some pure Hodge structure.

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References

  1. Arnold, V.I.; Gusein-Zade, S.M.; Varchenko, A.N.: Singularities of Differentiable Maps. Vol. II, Boston Basel Berlin: Birkhäuser (1988).

    Book  MATH  Google Scholar 

  2. Brieskorn, E.: Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscripta Math. 2, 103–161, (1970).

    Article  MathSciNet  MATH  Google Scholar 

  3. Clemens, H.; Griffiths, P.: The intermediate Jacobian of the cubic threefold. Ann. Math. 95, 281–356, (1972).

    Article  MathSciNet  MATH  Google Scholar 

  4. Deligne, P.: Equations différentielles à points singuliers réguliers. Lecture Notes Math. vol. 163, Berlin Heidelberg New York: Springer (1970).

    MATH  Google Scholar 

  5. Häuser, H; Müller, G.: The cancellation property for direct products of analytic spaces. Math. Ann. 286, 209–223, (1990).

    Article  MathSciNet  Google Scholar 

  6. Hertling, C.: Analytische Invarianten bei den unimodularen und bi-modularen Hyperflächensingularitäten. Dissertation. Bonner Math. Schriften 250, Bonn (1992).

    Google Scholar 

  7. Hertling, C.: Ein Torellisatz für die unimodalen und bimodularen Hyper flächensingularitäten. Math. Ann. 302, 359–394, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  8. Hertling, C.: Classifying spaces and moduli spaces for polarized mixed Hodge structures and for Brieskorn lattices. Preprint.

    Google Scholar 

  9. Malgrange, B.: Intégrales asymptotiques et monodromie. Ann. Sci. École Norm. Sup. 7, 405–430 (1974).

    MathSciNet  MATH  Google Scholar 

  10. Mather, J.; Yau, St.S.-T.: Classification of isolated hypersurface singularities by their moduli algebras. Invent. Math. 69, 243–252, (1982).

    Article  MathSciNet  MATH  Google Scholar 

  11. Milnor, J.: Singular points of complex hypersurfaces. Ann. Math. Stud, vol. 61, Princeton University Press (1968).

    Google Scholar 

  12. Pham, F.: Singularités des systèmes différentielles de Gauss-Manin. Prog. Math. 2, Boston Basel Stuttgart: Birkhäuser (1979).

    Google Scholar 

  13. Pham, F.: Structure de Hodge mixte associée à point critique isolé. Astérisque 101–102, 268–285, (1983).

    MathSciNet  Google Scholar 

  14. Saito, K.: The higher residue pairings K (k)F for a family of hypersurf ace singular points. In: Singularities, Proc. of symp. in pure math. 40.2, 441–463, (1983).

    Google Scholar 

  15. Saito, K.: Period mapping associated to a primitive form. Publ. Res. Inst. Math. Sci. Kyoto Univ. 19, 1231–1264, (1983).

    Article  MATH  Google Scholar 

  16. Saito, M.: On the structure of Brieskorn lattices. Ann. Inst. Fourier Grenoble 39, 27–72, (1989).

    Article  MathSciNet  MATH  Google Scholar 

  17. Saito, M.: Period mapping via Brieskorn modules. Bull. Soc. math. France 119, 141–171, (1991).

    MathSciNet  MATH  Google Scholar 

  18. Scherk, J.; Steenbrink, J.H.M.: On the mixed Hodge structure on the cohomology of the Milnor fibre. Math. Ann. 271, 641–665, (1985).

    Article  MathSciNet  MATH  Google Scholar 

  19. Scherk, J.: On the monodromy theorem for isolated hypersurface singularities. Invent. Math. 58, 289–301, (1980).

    Article  MathSciNet  MATH  Google Scholar 

  20. Schmid, W.: Variation of Hodge structure: The singularities of the period mapping. Invent. Math. 22, 211–319, (1973).

    Article  MathSciNet  MATH  Google Scholar 

  21. Sebastiani, M.; Thorn, R.: Un résultat sur la monodromie. Invent. Math. 13, 90–96, (1971).

    Article  MathSciNet  MATH  Google Scholar 

  22. Steenbrink, J.H.M.: Mixed Hodge structure on the vanishing cohomology. In: Real and complex singularities, Oslo (1976), Holm, P.(ed.). Arphen aan den Rijn: Sijthoff and Noordhoff, pp. 525–562, (1977).

    Chapter  Google Scholar 

  23. Tjurin, A.N.: The geometry of the Fano surface of a nonsingular cubic F ⊂ ℙ4 and Torell’s theorem for Fano surfaces and cubics. Math. USSR Izvestiya 5, 517–546, (1971).

    Article  Google Scholar 

  24. Varchenko, A.N.: The asymptotics of holomorphic forms determine a mixed Hodge structure. Sov. Math. Dokl. 22, 772–775, (1980).

    MATH  Google Scholar 

  25. Varchenko, A.N.: The complex singular index does not change along the stratum μ =constant. Functional Anal. Appl. 16, 1–9, (1982).

    Article  MathSciNet  MATH  Google Scholar 

  26. Varchenko, A.N.: A lower bound for the codimension of the stratum μ =constant in terms of the mixed Hodge structure. Moscow Univ. Math. Bull. 37, 30–33, (1982).

    MathSciNet  MATH  Google Scholar 

  27. Varchenko, A.N.: On the local residue and the intersection form on the vanishing cohomology. Math. USSR Izvestiya 26, 31–52, (1986).

    Article  MATH  Google Scholar 

  28. Washington, L.C.: Introduction to cyclotomic fields. New York: Springer (1982).

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut, Universität Bonn, Beringstraße 3, D - 53115, Bonn, Germany

    Claus Hertling

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  1. Claus Hertling
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Editor information

Editors and Affiliations

  1. Department of Geometry and Topology, Steklov Mathematical Institute, 8, Gubkina Stree, 117966, Moscow GSP-1, Russia

    V. I. Arnold

  2. CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, Postfach 3049, F-75775, Paris Cedex 16e, France

    V. I. Arnold

  3. Fachbereich Mathematik, Universität Kaiserslautern, D-67653, Kaiserslautern, Germany

    G.-M. Greuel

  4. Subfaculteit Wiskunde, Katholieke Universiteit Nijmegen Toernooiveld, NL-6525 ED, Nijmegen, The Netherlands

    J. H. M. Steenbrink

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Hertling, C. (1998). Brieskorn Lattices and Torelli Type Theorems for Cubics in ℙ3 and for Brieskorn-Pham Singularities with Coprime Exponents. In: Arnold, V.I., Greuel, GM., Steenbrink, J.H.M. (eds) Singularities. Progress in Mathematics, vol 162. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8770-0_9

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  • DOI: https://doi.org/10.1007/978-3-0348-8770-0_9

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9767-9

  • Online ISBN: 978-3-0348-8770-0

  • eBook Packages: Springer Book Archive

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