Skip to main content
SpringerLink
Log in

Rigid and Complete Intersection Lagrangian Singularities

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract.

In this article we prove a rigidity theorem for lagrangian singularities by studying the local cohomology of the lagrangian de Rham complex that was introduced in [SvS03]. The result can be applied to show the rigidity of all open swallowtails of dimension ≥ 2. In the case of lagrangian complete intersection singularities the lagrangian de Rham complex turns out to be perverse. We also show that lagrangian complete intersections in dimension greater than two cannot be regular in codimension one.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnol’d, V.I.: Lagrangian manifolds with singularities asymptotic rays and the open swallowtail. Funct. Anal. Appl. 15, 235–246 (1982)

    MATH  Google Scholar 

  2. Arnol’d, V.I.: Singularities of systems of rays. Russ. Math. Surv. 38 (2), 87–176 (1983)

    MATH  Google Scholar 

  3. Brieskorn, E.: Rationale Singularitäten komplexer Flächen. Invent. Math. 4, 336–358 (1968)

    MATH  Google Scholar 

  4. Diaz, S., Harris, J.: Ideals associated to deformations of singular plane curves. Trans. Am. Math. Soc. 309 (2), 433–468 (1988)

    MATH  Google Scholar 

  5. Fantechi, B., Göttsche, L., van Straten, D.: Euler number of the compactified Jacobian and multiplicity of rational curves. J. Algebr. Geom. 8 (1), 115–133 (1999)

    MATH  Google Scholar 

  6. Flenner, H.: Extendability of differential forms on non-isolated singularities. Invent. Math. 94 (2), 317–326 (1988)

    MATH  Google Scholar 

  7. Givental, A.B.: Singular Lagrangian manifolds and their Lagrangian mappings. Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Novejshie Dostizh. 33, 55–112 (1988), English translation: J. Soviet Math. 52.4, 3246–3278 (1990)

    MATH  Google Scholar 

  8. Givental, A.B.: Whitney singularities of solutions of partial differential equations. J. Geom. Phys. 15 (4), 353–368 (1995)

    Article  MATH  Google Scholar 

  9. Greuel, G.-M.: Der Gauß-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten. Math. Ann. 214, 235–266 (1975)

    MATH  Google Scholar 

  10. Grothendieck, A.: Local cohomology. A seminar given by A. Grothendieck Harvard University Fall 1961. Notes by R. Hartshorne. Lecture Notes in Mathematics, vol. 41, Springer-Verlag, 1967

  11. Hertling, C.: Frobenius Manifolds and Moduli Spaces for Singularities. Cambridge Tracts in Mathematics, vol. 151, Cambridge University Press, 2002

  12. Hochster, M.: The Zariski-Lipman conjecture for homogeneous complete intersections. Proc. Am. Math. Soc. 49, 261–262 (1975)

    MATH  Google Scholar 

  13. Hochster, M.: The Zariski-Lipman conjecture in the graded case. J. Algebra 47, 411–424 (1977)

    MATH  Google Scholar 

  14. Looijenga, E.J.N.: Isolated singular points on complete intersections. Lecture Note Series, vol. 77, Cambridge University Press, 1984

  15. Matsumura, H.: Commutative ring theory. Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, 1989

  16. Platte, E.: Ein elementarer Beweis des Zariski-Lipman-Problems für graduierte analytische Algebren. Arch. Math. 31, 143–145 (1978)

    MathSciNet  MATH  Google Scholar 

  17. Schlessinger, M.: Rigidity of quotient singularities. Invent. Math. 14, 17–26 (1971)

    MATH  Google Scholar 

  18. Sevenheck, C.: Lagrangian singularities. Ph.D. thesis. Johannes-Gutenberg-Universität Mainz, 2003, available at http://www.dma.ens.fr/∼sevenhec, p. 177

  19. Straten, D.V., Steenbrink, J.: Extendability of holomorphic differential forms near isolated hypersurface singularities. Abh. Math. Semin. Univ. Hamb. 55, 97–109 (1985)

    MATH  Google Scholar 

  20. Sevenheck, C., van Straten D.: Deformation of singular lagrangian subvarieties. Mathematische Annalen 327 (1), 79–102 (2003)

    Article  MATH  Google Scholar 

  21. Teissier, B.: Résolution simultanee. I. II. Seminaire sur les singularites des surfaces. Centre de Mathématiques de l’Ecole Polytechnique Palaiseau 1976–1977, Lecture Notes in Mathematics, vol. 777 Springer-Verlag, 1980

  22. Varchenko, A.N., Givental’, A.B.: Mapping of periods and intersection form. Funct. Anal. Appl. 16 83–93 (1982)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ecole Normale Supérieure, Département de mathématiques et applications, 45, rue d’Ulm, 75230, Paris cedex 05, France

    Christian Sevenheck

  2. FB Mathematik, Johannes-Gutenberg-Universität Mainz, FB Mathematik und Informatik, 55099, Mainz, Germany

    Duco van Straten

Authors
  1. Christian Sevenheck
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Duco van Straten
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Sevenheck.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sevenheck, C., van Straten, D. Rigid and Complete Intersection Lagrangian Singularities. manuscripta math. 114, 197–209 (2004). https://doi.org/10.1007/s00229-004-0456-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-004-0456-y

Keywords

Search

Navigation