Exotic involutions on irreducible 4-manifolds from Real Seiberg--Witten theory

Abstract

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 $X/⟨c⟩$. 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 ${\mathbb{CP}}^4$.

arXiv comment24 pages, no figures, comments are welcome