Symplectic symmetries of a hypothetical exotic $CP^2$: a potential construction or a nonexistence proof

Abstract

This paper is primarily concerned with the existence of a symplectic $4$-manifold, homeomorphic but not diffeomorphic to $CP^2$, which admits a nontrivial finite order symplectomorphism. We reduced the existence of such an exotic $CP^2$ to a question in contact geometry. More concretely, we showed that there are precisely $12$ plumbed manifolds, each equipped with a canonical contact structure associated to the plumbing, such that an exotic $CP^2$ with a nontrivial finite order symplectomorphism exists if and only if one of the $12$ contact plumbed manifolds admits a certain specific symplectic filling. As the main technical result, we completely determined the order and local representations of a pseudofree symplectic $Z_p$-action for an odd prime $p$ on a rational homology $CP^2$ with positive canonical class. Our result suggests that potentially, there are symplectic $Z_p$-actions which are not finite automorphisms of a fake projective plane. Moreover, we introduced a construction, which, under the minimal assumptions on the symplectic fillings of the $12$ contact plumbed manifolds, produces a symplectic $4$-manifold on the Bogomolov-Miyaoka-Yau line via a reverse-engineering process.