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The dihedral group $ \mathcal D_5$ as a group of symplectic automorphisms on K3 surfaces

Author: Alice Garbagnati
Journal: Proc. Amer. Math. Soc. 139 (2011), 2045-2055
MSC (2010): Primary 14J28, 14J50
Published electronically: January 11, 2011
MathSciNet review: 2775382
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Abstract: We prove that if a K3 surface $ X$ admits $ \mathbb{Z}/5\mathbb{Z}$ as a group of symplectic automorphisms, then it actually admits $ \mathcal{D}_5$ as a group of symplectic automorphisms. The orthogonal complement to the $ \mathcal{D}_5$-invariants in the second cohomology group of $ X$ is a rank 16 lattice, $ L$. It is known that $ L$ does not depend on $ X$: we prove that it is isometric to a lattice recently described by R. L. Griess Jr. and C. H. Lam. We also give an elementary construction of $ L$.

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Additional Information

Alice Garbagnati
Affiliation: Dipartimento di Matematica, Università di Milano, via Saldini 50, I-20133 Milano, Italia

Keywords: K3 surfaces, symplectic automorphisms, dihedral groups, lattices.
Received by editor(s): August 18, 2009
Received by editor(s) in revised form: February 5, 2010, June 3, 2010, and June 15, 2010
Published electronically: January 11, 2011
Communicated by: Ted Chinburg
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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