Symplectic symmetries of 4-manifolds

Details Symplectic symmetries of 4-manifolds Symplectic symmetries of 4-manifolds

2007-09-11

arxiv.org/abs/0709.1708v1

Topology, Vol. 46, No.2 (2007), 103-128

Mathematics

Geometric Topology

28 pages, 1 figure, 1 appendix, published

Scientific paper

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.

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