Abstract
For a very general complex projective K3 surface S and a smooth projective surface A with trivial canonical class, we prove that there is no dominant rational map A → S, which is not an isomorphism.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Amerik, M. Rovinsky and A. Van de Ven, A boundedness theorem for morphisms between threefolds, Université de Grenoble. Annales de l’Institut Fourier 49 (1999), 405–415.
A. Beauville, Endomorphisms of hypersurfaces and other manifolds, International Mathematics Research Notices 2001 (2001), 53–58.
S. Boissière, A. Sarti and D. C. Veniani, On prime degree isogenies between K3 surfaces, Rendiconti del Circolo Matematico di Palermo 66 (2017), 3–18.
X. Chen, Self rational maps of K3 surfaces, https://arxiv.org/abs/1008.1619.
T. Dedieu, Severi varieties and self-rational maps of K3 surfaces, International Journal of Mathematics 20 (2009), 1455–1477.
K. Hulek and M. Schütt, Enriques surfaces and Jacobian elliptic K3 surfaces, Mathematische Zeitschrift 268 (2011), 1025–1056.
D. Huybrechts, Lectures on K3 Surfaces, Cambridge Studies in Advanced Mathematics, Vol. 158, Cambridge University Press, Cambridge, 2016.
I. Karzhemanov, On endomorphisms of hypersurfaces, Kodai Mathematical Journal 40 (2017), 615–624.
C. Liedtke, Lectures on supersingular K3 surfaces and the crystalline Torelli theorem, in K3 Surfaces and Their Moduli, Progress in Mathematics, Vol. 315, Birkhäuser/Springer, Cham, 2016, pp. 171–235.
C. Liedtke, Supersingular K3 surfaces are unirational, Inventiones Mathematicae 200 (2015), 979–1014.
S. Ma, On K3 surfaces which dominate Kummer surfaces, Proceedings of the American Mathematical Society 141 (2013), 131–137.
D. R. Morrison, On K3 surfaces with large Picard number, Inventiones Mathematicae 75 (1984), 105–121.
D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Institut de Hautes Études Scientifiques. Publications Mathématiques 9 (1961), 5–22.
V. V. Nikulin, Finite groups of automorphisms of Kählerian K3 surfaces, Trudy Moskovskogo Matematicheskogo Obshchestva 38 (1979), 75–137.
V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 43 (1979), 111–177, 238.
V. V. Nikulin, On rational maps between K3 surfaces, in Constantin Carathéodory: an International Tribute. Vols. I, II, World Scientific, Teaneck, NJ, 1991, pp. 964–995.
T. Shioda, Kummer sandwich theorem of certain elliptic K3 surfaces, Japan Academy. Proceedings. Series A. Mathematical Sciences 82 (2006), 137–140.
T. Shioda and H. Inose, On singular K3 surfaces, in Complex Analysis and Algebraic Geometry, Iwanami Shoten, Tokyo, 1977, pp. 119–136.
C. Voisin, A geometric application of Nori’s connectivity theorem, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 3 (2004), 637–656.
Acknowledgments
We would like to thank S. Galkin for introducing us to the subject, treated in Theorem 1.1, and F. Bogomolov, A. Bondal, I. Dolgachev, V. Nikulin, Y. Zarhin, I. Zhdanovskiy for their interest and valuable comments. We are also grateful to an anonymous referee whose remarks and suggestions have improved the exposition of our paper. This work was supported by the Priority 2030 Strategic Academic Leadership Program and by the HSE University Basic Research Program.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Karzhemanov, I., Konovalov, G. Rational maps and K3 surfaces. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2656-3
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s11856-024-2656-3