An adjunction inequality for Real embedded surfaces

Abstract

A Real structure on a $4$-manifold $X$ is an orientation preserving smooth involution $\ddot{I}ƒ$. We say that an embedded surface $\hat{I}\pounds \subset X$ is Real if $\ddot{I}ƒ$ maps $\hat{I}\pounds$ 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.

25 pages, minor corrections. To appear in Int. Math. Res. Not