Global classification of curves on the symplectic plane

In [17], we considered the local classification of plane curves on the symplectic plane. In particular, we introduced the number “symplectic defect”, which represents the difference of two natural equivalence relations on plane curves, the equivalence by diffeomorphisms and that by symplectomorphisms. For an immersion, two equivalence relations coincide, so the symplectic defect is null. For complicated singularities, the symplectic defects turn out to be positive. In this paper we consider the global symplectic classification problem. First we give the exact classification result under symplectomorphisms, for the case of generic plane curves, namely immersions with transverse self-intersections. Then, for a given diffeomorphism class of a generic plane curve, the set of symplectic classes form the symplectic moduli space which we completely describe by its global topological term (Theorem 1.1). In the general plane curves with singularities, the difference between symplectomorphism and diffeomorphism classifications is clearly described by local symplectic moduli spaces of singularities and a global topological term. Thus, up to the classification by diffeomorphisms, the global problem is reduced to the local classification problem (Theorem 1.4). We introduce the symplectic moduli space of a global plane curve and the local symplectic moduli space of a plane curve singularity as quotients of mapping spaces, and we endow them with differentiable structures in a natural way. Actually we treat labelled plane curves and labelled symplectic moduli spaces. For a plane curve, we label all compact domains surrounded by it and all singular points, and consider the classification problem of plane curves isotopic to the given plane curve by symplectomorphisms preserving the labelling.