Invariants of real symplectic four-manifolds out of reducible and cuspidal curves

We construct invariants under deformation of real symplectic four-manifolds. These invariants are obtained by counting three different kinds of real rational J-holomorphic curves which realize a given homology class and pass through a given real configuration of (the appropriate number of) points. These curves are cuspidal curves, reducible curves and curves with a prescribed tangent line at some real point of the configuration. They are counted with respect to some sign defined by the parity of their number of isolated real double points and in the case of reducible curves, with respect to some mutiplicity. In the case of the complex projective plane equipped with its standard symplectic form and real structure, these invariants coincide with the ones previously constructed in [11]. This leads to a relation between the count of real rational J-holomorphic curves done in [11] and the count of real rational reducible J-holomorphic curves presented here.