Towards relative invariants of real symplectic four-manifolds

Abstract

Let (X, ω, c X ) be a real symplectic four-manifold with real part \(\mathbb{R}X\). Let \(L \subset \mathbb{R}X\) be a smooth curve such that \([L] = 0 \in H_1 (\mathbb{R}X;\mathbb{Z}/2\mathbb{Z}).\) We construct invariants under deformation of the quadruple (X, ω, c X , 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 [W2] and the count of reducible real rational curves done in [W3]. Finally, we show how these techniques also allow us 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.