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Finiteness and quasi-simplicity for symmetric \(K3\)-surfaces. (English) Zbl 1073.14053

The authors consider the finite groups of homomorphic and anti-holomorphic transformations of complex (not necessarily algebraic) \(K3\) surfaces and obtain the following results:
(Finiteness) The number of equivariant deformation classes of \(K3\) surfaces with faithful Klein actions of finite groups is finite.
(Weak quasi-simplicity) Two \(K3\) surfaces with a finite group of Klein actions of the same homological type are equivariantly deformation equivalent up to the conjugation of the complex structure, provided that either the action is holomorphic, or the representation of the group in \(U(1)\), acting on \(H^{2,0}\simeq{\mathbb C}\), is conjugation invariant.
If this representation is trivial then the surfaces are equivariantly deformation equivalent. This can be viewed as an equivariant Torelli theorem for \(K3\) surfaces. Similar results are obtained for 2-dimensional complex tori. In addition, the authors give two examples of \(K3\) surfaces showing that the “up to the conjugation of the complex structure” condition in the weak quasi-simplicity theorem cannot be removed.

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14P25 Topology of real algebraic varieties
32G05 Deformations of complex structures
57S17 Finite transformation groups
14J50 Automorphisms of surfaces and higher-dimensional varieties
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