Exotic involutions on irreducible 4-manifolds from Real Seiberg--Witten theory
Using Real Seiberg--Witten theory, Miyazawa constructed infinitely many exotic involutions on ${\overline{\mathbb{CP}}{}^2}$, and Baraglia extended this construction to $X\#{\overline{\mathbb{CP}}{}^2}$ for a broad class of $4$-manifolds $X$ equipped with an involution $c$. We show that, under suitable Real Seiberg--Witten hypotheses, the ${\overline{\mathbb{CP}}{}^2}$ summand can be omitted when $X$ is non-spin and the complement of the fixed locus is simply-connected. Our construction also yields infinite families of topologically isotopic but smoothly non-isotopic embeddings of the branch surface in the quotient $X/\langle c\rangle$. As applications, we construct anti-holomorphic involutions on irreducible, simply-connected projective surfaces of general type with infinitely many exotic copies, none of which is anti-symplectic with respect to any orientation-compatible symplectic form. These surfaces include cyclic branched covers of $\mathbb{CP}^2$ and complete intersections of multidegree $(2,4k)$ in $\mathbb{CP}^4$.