Duke Mathematical Journal

Finiteness and quasi-simplicity for symmetric K3 -surfaces

Alex Degtyarev, Ilia Itenberg, and Viatcheslav Kharlamov

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Abstract

We compare the smooth and deformation equivalence of actions of finite groups on K3 -surfaces by holomorphic and antiholomorphic transformations. We prove that the number of deformation classes is finite and, in a number of cases, establish the expected coincidence of the two equivalence relations. More precisely, in these cases we show that an action is determined by the induced action in the homology. On the other hand, we construct two examples to show first that, in general, the homological type of an action does not even determine its topological type, and second that K3 -surfaces X and X ¯ with the same Klein action do not need to be equivariantly deformation equivalent even if the induced action on H 2,0 (X) is real, that is, reduces to multiplication by ±1 .

Article information

Source
Duke Math. J., Volume 122, Number 1 (2004), 1-49.

Dates
First available in Project Euclid: 24 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1080137201

Digital Object Identifier
doi:10.1215/S0012-7094-04-12211-8

Mathematical Reviews number (MathSciNet)
MR2046806

Zentralblatt MATH identifier
1073.14053

Subjects
Primary: 14J28: $K3$ surfaces and Enriques surfaces 14J50: Automorphisms of surfaces and higher-dimensional varieties
Secondary: 14P25: Topology of real algebraic varieties 32G05: Deformations of complex structures [See also 13D10, 16S80, 58H10, 58H15] 57S17: Finite transformation groups

Citation

Degtyarev, Alex; Itenberg, Ilia; Kharlamov, Viatcheslav. Finiteness and quasi-simplicity for symmetric $K3$ -surfaces. Duke Math. J. 122 (2004), no. 1, 1--49. doi:10.1215/S0012-7094-04-12211-8. https://projecteuclid.org/euclid.dmj/1080137201


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