An adjunction inequality for Real embedded surfaces

Abstract A Real structure on a $4$-manifold $X$ is an orientation preserving smooth involution $\sigma $. We say that an embedded surface $\Sigma \subset X$ is Real if $\sigma $ maps $\Sigma $ to itself orientation reversingly. We prove that a cohomology class $u \in H^{2}(X ; \mathbb{Z})$ can be represented by a Real embedded surface if and only if $u$ can be lifted to a class in equivariant cohomology $H^{2}_{\mathbb{Z}_{2}}(X ; \mathbb{Z}_{-})$. We prove that if the Real Seiberg–Witten invariants of $X$ are non-zero then the genus of Real embedded surfaces in $X$ satisfy an adjunction inequality. We prove two versions of the adjunction inequality, one for non-negative self-intersection and one for arbitrary self-intersection. We show with examples that the minimal genus of Real embedded surfaces can be larger than the minimal genus of arbitrary embedded surfaces.