On local classification of singular symplectic forms.

Wojciech Domitrz Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland, e-mail: domitrz@mini.pw.edu.pl

April 19, 2002

Let ω be a closed 2-form on R^{2n}, ω is called the singular form if ω^{n} = fΩ, where Ω is the volume form and f is a smooth function f : R^{2n} → R such that Σ_{2} = f^{−1}(0) ≠ ∅. In [3], [5] and [2] it was shown that there are five generic types of singularities for germs of 2-forms on R^{4}. All these generic types, among which, one is unstable and has moduli parameters, are determined by the geometry of the pullback i^{∗}ω to the hypersurface of degeneration i : Σ_{2} → R^{4}.

If germs of singular symplectic forms are equivalent then their pullbacks to the hypersurfaces of degeneration are equivalent. We consider the following problem :

Does the pullback of the singular symplectic form to its hypersurface of degeneration determine the equivalence class of the form?

References

[1] W. Domitrz, S. Janeczko and Z. Pasternak-Winiarski Geometry and representation of the singular symplectic forms, preprint.

[2] M. Golubitsky and D. Tischler, An example of moduli for singular symplectic forms, Inventiones Math., 38 (1977), 219-225.

[3] J. Martinet, Sur les singularités des formes differentielles, Ann. Inst. Fourier (Grenoble), 20 (1970), 95-178.

[4] J. Moser, On volume elements on manifold, Trans. Amer. Math. Soc., 120 (1965), 280-296.

[5] R. Roussarie, Modèles locaux de champs et de formes, Astérisque, 30, (1975), 1-181.