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St. Petersburg Mathematical Journal

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Classification of simple multigerms of curves in a space with symplectic structure


Author: P. A. Kolgushkin
Translated by: N. Yu. Netsvetaev
Original publication: Algebra i Analiz, tom 15 (2003), nomer 1.
Journal: St. Petersburg Math. J. 15 (2004), 103-126
MSC (2000): Primary 57R45
Published electronically: December 31, 2003
MathSciNet review: 1979720
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Abstract | References | Similar Articles | Additional Information

Abstract: A classification of stably simple germs of curves (both reducible and irreducible) in the complex space equipped with a symplectic structure is obtained. This classification extends the result by V. I. Arnol'd of 1999, which described the $A_{2k}$ singularities in the symplectic complex space. The proofs involve the homotopy method and the Darboux-Givental theorem.


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  • [1] V. I. Arnol′d, Simple singularities of curves, Tr. Mat. Inst. Steklova 226 (1999), no. Mat. Fiz. Probl. Kvantovoi Teor. Polya, 27–35 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 3 (226) (1999), 20–28. MR 1782550
  • [2] V. I. Arnold, First steps of local symplectic algebra, Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, vol. 194, Amer. Math. Soc., Providence, RI, 1999, pp. 1–8. MR 1729356
  • [3] V. I. Arnol′d and A. B. Givental′, Symplectic geometry, Current problems in mathematics. Fundamental directions, Vol. 4, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 5–139, 291 (Russian). MR 842908
  • [4] J. W. Bruce and T. J. Gaffney, Simple singularities of mappings 𝐶,0→𝐶²,0, J. London Math. Soc. (2) 26 (1982), no. 3, 465–474. MR 684560, 10.1112/jlms/s2-26.3.465
  • [5] B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern geometry—methods and applications. Part I, 2nd ed., Graduate Texts in Mathematics, vol. 93, Springer-Verlag, New York, 1992. The geometry of surfaces, transformation groups, and fields; Translated from the Russian by Robert G. Burns. MR 1138462
  • [6] C. G. Gibson and C. A. Hobbs, Simple singularities of space curves, Math. Proc. Cambridge Philos. Soc. 113 (1993), no. 2, 297–310. MR 1198413, 10.1017/S0305004100075976
  • [7] P. A. Kolgushkin and R. R. Sadykov, Classification of simple multigerms of curves, Uspekhi Mat. Nauk 56 (2001), no. 6, 153-154; English transl., Russian Math. Surveys 56 (2001), no. 6, 1166-1167.
  • [8] -, Simple singularities of multigerms of curves, Rev. Mat. Comput. 14 (2001), no. 2, 311-344; arxiv.org/abs/math.AG/0012040.
  • [9] John N. Mather, Stability of 𝐶^{∞} mappings. IV. Classification of stable germs by 𝑅-algebras, Inst. Hautes Études Sci. Publ. Math. 37 (1969), 223–248. MR 0275460

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Additional Information

P. A. Kolgushkin
Affiliation: Moscow State University, Mechanics and Mathematics Department, Moscow 119899, Russia
Email: kolgush@mccme.ru

DOI: https://doi.org/10.1090/S1061-0022-03-00804-5
Keywords: Symplectomorphism, stably simple singularity, multigerm
Received by editor(s): January 29, 2002
Published electronically: December 31, 2003
Additional Notes: Partly supported by RFBR (grant no. 01-01-00739) and by NWD-RFBR (grant no. 047.008.005).
Article copyright: © Copyright 2003 American Mathematical Society