Mathematische Annalen

, Volume 345, Issue 4, pp 783–817 | Cite as

Volume preserving subgroups of \({{\mathcal A}}\) and \({{\mathcal K}}\) and singularities in unimodular geometry

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Abstract

For a germ of a smooth map f from \({{\mathbb K}^n}\) to \({{\mathbb K}^p}\) and a subgroup \({{{G}_{\Omega _q}}}\) of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form Ω q in \({{\mathbb K}^q}\) (q = n or p) we study the \({{{G}_{\Omega _q}}}\) -moduli space of f that parameterizes the \({{{G}_{\Omega _q}}}\) -orbits inside the G-orbit of f. We find, for example, that this moduli space vanishes for \({{{G}_{\Omega _q}} ={{\mathcal A}_{\Omega _p}}}\) and \({{\mathcal A}}\)-stable maps f and for \({{{G}_{\Omega _q}} ={{\mathcal K}_{\Omega _n}}}\) and \({{\mathcal K}}\)-simple maps f. On the other hand, there are \({{\mathcal A}}\)-stable maps f with infinite-dimensional \({{{\mathcal A}_{\Omega _n}}}\) -moduli space.

Mathematics Subject Classification (2000)

32S05 32S30 58K40 
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References

  1. 1.
    Arnol’d V.I.: Remarks on Poisson structures on a plane and on other powers of volume elements. J. Sov. Math. 47, 2509–2516 (1989)MATHCrossRefGoogle Scholar
  2. 2.
    Arnol’d V.I.: Simple singularities of curves. Proc. Steklov Inst. Math. 226, 20–28 (1999)Google Scholar
  3. 3.
    Arnol’d, V.I.: First Steps of Local Symplectic Algebra. Advances in Soviet Mathematics, D. Fuchs birthday volume. American Mathematical Society, Providence (1999)Google Scholar
  4. 4.
    Bloom T., Herrera M.: De Rham cohomology of an analytic space. Invent. Math. 7, 275–296 (1969)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Brieskorn E.: Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscr. Math. 2, 103–161 (1970)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bruce J.W., Gaffney T.: Simple singularities of maps \({({\mathbb C}, 0) \rightarrow ({\mathbb C}^2, 0)}\). J. Lond. Math. Soc. 26, 464–474 (1982)MathSciNetGoogle Scholar
  7. 7.
    Bruce J.W., du Plessis A.A., Wilson L.C.: Discriminants and liftable vector fields. J. Algebraic Geom. 3, 725–753 (1994)MATHMathSciNetGoogle Scholar
  8. 8.
    Colin de Verdière Y.: Singular Lagrangian manifolds and semi-classical analysis. Duke Math. J. 116, 263–298 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Colin de Verdière Y., Vey J.: Le lemme de Morse isochore. Topology 18, 283–293 (1979)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cooper T., Mond D., Wik Atique R.: Vanishing topology of codimension 1 multi-germs over \({{\mathbb R}\,{\rm and}\,{\mathbb C}}\). Compositio Math. 131, 121–160 (2002)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Damon, J.: The unfolding and determinacy results for subgroups of \({{\mathcal A}\,{\rm and}\,{\mathcal K}}\). In: Proceedings of Symposia in Pure Mathematics, vol. 40, pp. 233–254, Part 1. American Mathematical Society, Providence (1983)Google Scholar
  12. 12.
    Damon, J.: The unfolding and determinacy results for subgroups of \({{\mathcal A}}\) and \({{\mathcal K}}\). Memoirs American Mathematical Society, vol. 50, no. 306. American Mathematical Society, Providence (1984)Google Scholar
  13. 13.
    Damon J., Mond D.: \({{\mathcal A}}\)-codimension and the vanishing topology of discriminants. Invent. Math. 106, 217–242 (1991)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dimca A., Gibson C.G.: Contact unimodular germs from the plane to the plane. Q. J. Math. Oxford 34, 281–295 (1983)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Dimca A., Gibson C.G.: Classification of equidimensional contact unimodular map germs. Math. Scand. 56, 15–28 (1985)MATHMathSciNetGoogle Scholar
  16. 16.
    Domitrz, W.: Local symplectic algebra of quasi-homogeneous curves. Fundam. Math. (to appear)Google Scholar
  17. 17.
    Domitrz W., Janeczko S., Zhitomirskii M.: Relative Poincare lemma, contractibility, quasi-homogeneity and vector fields tangent to a singular variety. Ill. J. Math. 48, 803–835 (2004)MATHMathSciNetGoogle Scholar
  18. 18.
    Domitrz W., Janeczko S., Zhitomirskii M.: Symplectic singularities of varieties: the method of algebraic restrictions. J. Reine Angew. Math. 618, 197–235 (2008)MATHMathSciNetGoogle Scholar
  19. 19.
    Francoise J.-P.: Modèle local simultané d’une fonction et d’une forme de volume. Astèrisque S. M. F. 59–60, 119–130 (1978)Google Scholar
  20. 20.
    Francoise J.-P.: Réduction simultanée d’un croisement normal et d’un volume. Bol. Soc. Bras. Mat. 13, 79–83 (1982)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Gabrielov A.M.: Bifurcations, Dynkin diagrams and the moduli number for isolated singularities. Funct. Anal. Appl. 8, 7–13 (1974)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Garay M.D.: An isochore versal deformation theorem. Topology 43, 1081–1088 (2004)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Gibson C.G., Hobbs C.A.: Simple singularities of space curves. Math. Proc. Camb. Phil. Soc. 113, 297–310 (1993)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Giusti M.: Classification des singularités isolées d’intersections complètes simples. C. R. Acad. Sci. Paris. Sér. A-B 284, 167–169 (1977)MATHMathSciNetGoogle Scholar
  25. 25.
    Goryunov V.V.: Singularities of projections of complete intersections. J. Sov. Math. 27, 2785–2811 (1984)MATHCrossRefGoogle Scholar
  26. 26.
    Greuel G.-M.: Der Gauss–Manin–Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten. Math. Ann. 214, 235–266 (1975)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Greuel G.-M.: Dualität in der lokalen Kohomologie isolierter Singularitäten. Math. Ann. 250, 157–173 (1980)MATHMathSciNetGoogle Scholar
  28. 28.
    Houston, K., Kirk, N.P.: On the classification and geometry of corank 1 map-germs from three- to four-space. In: Singularity Theory, London Mathematical Society. Lecture Note Series, vol. 263, pp. 325–351. Cambridge University Press, Cambridge (1999)Google Scholar
  29. 29.
    Ishikawa G., Janeczko S.: Symplectic bifurcations of plane curves and isotropic liftings. Q. J. Math. Oxford Ser. 54(2), 73–102 (2003)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Klotz C., Pop O., Rieger J.H.: Real double points of deformations of \({{\mathcal A}}\)-simple map-germs from \({{\mathbb R}^n\,{\rm to}\,{\mathbb R}^{2n}}\). Math. Proc. Camb. Phil. Soc. 142, 341–363 (2007)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Kostov, V.P., Lando, S.K.: Versal deformations of powers of volume forms. In: Computational Algebraic Geometry (Nice, 1992). Progr. Math., vol. 109, pp. 143–162. Birkhäuser Boston, Boston (1993)Google Scholar
  32. 32.
    Lando, S.K.: Deformations of differential forms, Trudy Mat. Inst. Steklov., vol. 209, Osob. Gladkikh Otobrazh. s Dop. Strukt., pp. 167–199 (Russian) (1995)Google Scholar
  33. 33.
    Lando, S.K.: Singularities of the differential forms of the complex degree and their deformations. In: Singularity Theory (Trieste, 1991), pp. 289–305. World Scientific Publishing, River Edge (1995)Google Scholar
  34. 34.
    Lando S.K.: Normal forms of powers of volume forms. Funktsional. Anal. i Prilozhen 19, 78–79 (1985) (Russian)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Malgrange B.: Intégrales asymptotiques et monodromie. Ann. Sci. École Norm. Sup. 20, 147–176 (1974)MATHMathSciNetGoogle Scholar
  36. 36.
    Marar W.L., Tari F.: On the geometry of simple germs of corank 1 from \({{\mathbb R}^3\,{\rm to}\,{\mathbb R} ^3}\). Math. Proc. Camb. Philos. Soc. 119, 469–481 (1996)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Martinet, J.: Singularities of Smooth Functions and Maps. London Mathematical Society. Lecture Note Series, vol. 58. Cambridge University Press, Cambridge (1982)Google Scholar
  38. 38.
    Mather J.N.: Stability of C ∞ mappings, IV: classification of stable germs by \({{\mathbb R}}\)-algebras. Publ. Math. I.H.E.S. 37, 223–248 (1969)MATHMathSciNetGoogle Scholar
  39. 39.
    Mather J.N.: Stability of C ∞ mappings, V: transversality. Adv. Math. 4, 301–336 (1970)MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Mather, J.N.: Stability of C ∞ mappings, VI: the nice dimensions. In: Proceedings of the Liverpool Singularities Symposium I. Lecture Notes in Mathematics, vol. 192, pp. 207–253. Springer, Berlin (1970)Google Scholar
  41. 41.
    Mond D.: On the classification of germs of maps from \({{\mathbb R}^2\,{\rm to}\,{\mathbb R}^3}\). Proc. Lond. Math. Soc. 50, 333–369 (1985)MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Moser J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 280–296 (1965)CrossRefGoogle Scholar
  43. 43.
    Reiffen H.-J.: Das Lemma von Poincaré für holomorphe Differentialformen auf komplexen Räumen. Math. Z. 101, 269–284 (1967)MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Rieger J.H.: Families of maps from the plane to the plane. J. Lond. Math. Soc. 36, 351–369 (1987)MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Rieger J.H.: \({{\mathcal A}}\)-unimodal map-germs into the plane. Hokkaido Math. J. 33, 47–64 (2004)MATHMathSciNetGoogle Scholar
  46. 46.
    Rieger J.H., Ruas M.A.S.: Classification of \({{\mathcal A}}\)-simple germs from k n to k 2. Compositio Math. 79, 99–108 (1991)MATHMathSciNetGoogle Scholar
  47. 47.
    Sebastiani M.: Preuve d’une conjecture de Brieskorn. Manuscr. Math. 2, 301–308 (1970)MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Varchenko A.N.: Local classification of volume forms in the presence of a hypersurface. Funktsional. Anal. i Prilozhen 19(4), 23–31 (1985) (Russian)MathSciNetGoogle Scholar
  49. 49.
    Vey J.: Sur le lemme de Morse. Invent. Math. 40, 1–10 (1977)MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Wall C.T.C.: Finite determinacy of smooth map-germs. Bull. Lond. Math. Soc. 13, 481–539 (1981)MATHCrossRefGoogle Scholar
  51. 51.
    Wall, C.T.C.: Classification of unimodal isolated singularities of complete intersections. In: Proceedings of Symposia in Pure Mathematics, vol. 40, Part 2, pp. 625–640 (1983)Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland
  2. 2.Institut für MathematikUniversität HalleHalle (Saale)Germany

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