An adjunction inequality for Real embedded surfaces

A Real structure on a $4$-manifold $X$ is an orientation preserving smooth involution $\sigma$. We say that an embedded surface $\mathrm{\Sigma }\subset X$ is Real if $\sigma$ maps $\mathrm{\Sigma }$ 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 $H_{{\mathbb{Z}}_2}^2(X;{\mathbb{Z}}_{-})$. 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.