Symplectic symmetries of 4-manifolds

TL;DR

This paper investigates symplectic actions of finite groups on 4-manifolds, using equivariant Seiberg-Witten-Taubes theory to classify fixed-point sets and demonstrate the triviality of many such symmetries, especially on K3 surfaces.

Contribution

It introduces a novel application of equivariant Seiberg-Witten-Taubes theory to classify fixed-point structures of symplectic group actions on 4-manifolds, especially for prime order cyclic actions.

Findings

Complete description of fixed-point set structure for prime order symplectic cyclic actions.

Establishment of triviality of many symplectic symmetries on certain 4-manifolds.

Proof of triviality of homologically trivial symplectic symmetries on K3 surfaces.

Abstract

A study of symplectic actions of a finite group $G$ on smooth 4-manifolds is initiated. The central new idea is the use of $G$-equivariant Seiberg-Witten-Taubes theory in studying the structure of the fixed-point set of these symmetries. The main result in this paper is a complete description of the fixed-point set structure (and the action around it) of a symplectic cyclic action of prime order on a minimal symplectic 4-manifold with $c_1^2=0$. Comparison of this result with the case of locally linear topological actions is made. As an application of these considerations, the triviality of many such actions on a large class of 4-manifolds is established. In particular, we show the triviality of homologically trivial symplectic symmetries of a $K3$ surface (in analogy with holomorphic automorphisms). Various examples and comments illustrating our considerations are also included.