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On prime degree isogenies between K3 surfaces

Abstract

We classify prime order isogenies between algebraic K3 surfaces whose rational transcendental Hodge structures are not isometric. The morphisms of Hodge structures induced by these isogenies are correspondences by algebraic classes on the product fourfolds; however, they do not satisfy the hypothesis of the well-known Mukai–Nikulin theorem. As an application we describe isogenies obtained from K3 surfaces with an action of a symplectic automorphism of prime order.

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Acknowledgments

We warmly thank Bert van Geemen, Xavier Roulleau and Matthias Schütt for helpful discussions. Part of the work was done when the third named author was visiting the University of Poitiers, he warmly thanks this institution for the kind hospitality and the stimulating working atmosphere. He was supported by the Research Training Group GRK 1463 of the Leibniz University of Hannover. We thank the anonymous referee for useful remarks.

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Correspondence to Alessandra Sarti.

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Dedicated to Philippe Ellia on his sixtieth birthday.

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Boissière, S., Sarti, A. & Veniani, D.C. On prime degree isogenies between K3 surfaces. Rend. Circ. Mat. Palermo, II. Ser 66, 3–18 (2017). https://doi.org/10.1007/s12215-016-0270-x

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Keywords

  • K3 surfaces
  • Automorphisms of surfaces
  • Algebraic cycles
  • Hodge theory

Mathematics Subject Classification

  • 14J28
  • 14J50
  • 14C25
  • 14C30