M. Golubitsky and D. Tischler
An example of moduli for singular symplectic forms
Inventiones Math. 38 (1977) 219-225.
In [3] Martinet shows that there are four generic types of singularities for germs of closed C^ifnty 2-forms on 4-manifolds and then defines a notion of stability for these germs. The stability of the first singularity type is just the classical Darboux theorem for symplectic forms. Martinet proved the stability of the second type; while, more recently, Roussarie [6] has shown the stability of the third. In this paper we shall show that forms exhibiting this last type of singularity are unfortunately not stable. In fact, we show that near any generic Sigma_{2,2,1} singularity there is, at least, a one parameter family of moduli.
In Section 1 we briefly describe the various singularities. In Section 2 we will show how to reduce the problem of stability to one involving a contact structure on R^3 at 0. Section 3 contains the proof of instability.
Note: we assume that all functions, forms, vector fields, etc. are C^ifnty.