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Journal of Topology 2008 1(4):923-962; doi:10.1112/jtopol/jtn027
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© 2008 London Mathematical Society

Symmetries and exotic smooth structures on a K3 surface

Weimin Chen

Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA wchen@math.umass.edu

Slawomir Kwasik

Mathematics Department, Tulane University, New Orleans, LA 70118, USA kwasik@math.tulane.edu


  Abstract

Smooth and symplectic symmetries of an infinite family of distinct exotic K3 surfaces are studied, and a comparison with the corresponding symmetries of the standard K3 is made. The action on the K3 lattice induced by a smooth finite group action is shown to be strongly restricted, and, as a result, the nonsmoothability of actions induced by a holomorphic automorphism of prime order at least 7 is proved, and the nonexistence of smooth actions by several K3 groups is established (included among which is the binary tetrahedral group T24 that has the smallest order). Concerning symplectic symmetries, the fixed-point set structure of a symplectic cyclic action of prime order at least 5 is explicitly determined, provided that the action is homologically nontrivial.

Received August 20, 2007.

2000 Mathematics Subject Classification 57S15, 57R55 (primary), 57R17 (secondary)

The first author is supported in part by the NSF grant DMS-0603932.


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