Towards relative invariants of real symplectic 4-manifolds

Let (X,ω, cX ) be a real symplectic 4-manifold with real part RX. Let L ⊂ RX be a smooth curve such that [L] = 0 ∈ H1(RX;Z/2Z). We construct invariants under deformation of the quadruple (X,ω, cX , L) by counting the number of real rational J-holomorphic curves which realize a given homology class d, pass through an appropriate number of points and are tangent to L. As an application, we prove a relation between the count of real rational J-holomorphic curves done in [9] and the count of reducible real rational curves done in [10]. Finally, we show how these techniques also allow to extract an integer valued invariant from a classical problem of real enumerative geometry, namely about counting the number of real plane conics tangent to five given generic real conics. 1 Statement of the results Let (X,ω, cX ) be a real symplectic 4-manifold, that is a triple made of a smooth compact 4-manifold X, a symplectic form ω on X and an involution cX on X such that c ∗ Xω = −ω. The fixed point set of cX is called the real part of X and is denoted by RX. It is assumed to be non empty here so that it is a smooth lagrangian surface of (X,ω). We label its connected components by (RX)1, . . . , (RX)N . Let L ⊂ RX be a smooth curve which represents 0 in H1(RX;Z/2Z), and B ⊂ RX be a surface having L as a boundary.