Symmetric symplectic homotopy K3 surfaces

A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is shown that an effective action by various maximal symplectic K3 groups forces the corresponding homotopy K3 surface to be minimally exotic with respect to our measure. (However, the standard K3 is the only known example of such minimally exotic homotopy K3 surfaces.) The possible structure of a finite group of symplectic symmetries of a minimally exotic homotopy K3 surface is determined and future research directions are indicated.Comment: 30 pages, 1 figure, with a slightly changed title. Accepted for publication by the Journal of Topolog

Symplectic symmetries of 4-manifolds

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.Comment: 28 pages, 1 figure, 1 appendix, publishe