On the orders of periodic diffeomorphisms of 4-manifolds

Abstract

This paper initiated an investigation on the following question: Suppose that a smooth $4$-manifold does not admit any smooth circle actions. Does there exist a constant $C>0$ such that the manifold supports no smooth ${\mathbb{Z}}_p$-actions of prime order for $p>C$? We gave affirmative results to this question for the case of holomorphic and symplectic actions, with an interesting finding that the constant $C$ in the holomorphic case is topological in nature, while in the symplectic case it involves also the smooth structure of the manifold.