Borchers lattices and K3 surfaces of zero entropy

Publikation: Arbeitspapier/Preprint › Preprint

Autoren

  • Simon Brandhorst
  • Giacomo Mezzedimi

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OriginalspracheEnglisch
Seitenumfang30
Band2022
PublikationsstatusElektronisch veröffentlicht (E-Pub) - 17 Nov. 2022

Publikationsreihe

NameArXiv

Abstract

Let L be an even, hyperbolic lattice with infinite symmetry group. We call L a Borcherds lattice if it admits an isotropic vector with bounded inner product with all the simple (−2)-roots. We show that this is the case if and only if L has zero entropy, or equivalently if and only if all symmetries of L preserve some isotropic vector. We obtain a complete classification of Borcherds lattices, consisting of 194 lattices. In turn this provides a classification of hyperbolic lattices with virtually abelian symmetry group and rank ≥5. Finally, we apply these general results to the case of K3 surfaces. We obtain a classification of Picard lattices of K3 surfaces of zero entropy and infinite automorphism group, consisting of 193 lattices. In particular we show that all Kummer surfaces, all supersingular K3 surfaces and all K3 surfaces covering an Enriques surface (with one exception) admit an automorphism of positive entropy.

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Borchers lattices and K3 surfaces of zero entropy. / Brandhorst, Simon; Mezzedimi, Giacomo.
2022. (ArXiv).

Publikation: Arbeitspapier/Preprint › Preprint

Brandhorst, S., & Mezzedimi, G. (2022). Borchers lattices and K3 surfaces of zero entropy. (ArXiv). Vorabveröffentlichung online. https://arxiv.org/abs/2211.09600
Brandhorst S, Mezzedimi G. Borchers lattices and K3 surfaces of zero entropy. 2022 Nov 17. (ArXiv). Epub 2022 Nov 17.
Brandhorst, Simon ; Mezzedimi, Giacomo. / Borchers lattices and K3 surfaces of zero entropy. 2022. (ArXiv).
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N2 - Let L be an even, hyperbolic lattice with infinite symmetry group. We call L a Borcherds lattice if it admits an isotropic vector with bounded inner product with all the simple (−2)-roots. We show that this is the case if and only if L has zero entropy, or equivalently if and only if all symmetries of L preserve some isotropic vector. We obtain a complete classification of Borcherds lattices, consisting of 194 lattices. In turn this provides a classification of hyperbolic lattices with virtually abelian symmetry group and rank ≥5. Finally, we apply these general results to the case of K3 surfaces. We obtain a classification of Picard lattices of K3 surfaces of zero entropy and infinite automorphism group, consisting of 193 lattices. In particular we show that all Kummer surfaces, all supersingular K3 surfaces and all K3 surfaces covering an Enriques surface (with one exception) admit an automorphism of positive entropy.

AB - Let L be an even, hyperbolic lattice with infinite symmetry group. We call L a Borcherds lattice if it admits an isotropic vector with bounded inner product with all the simple (−2)-roots. We show that this is the case if and only if L has zero entropy, or equivalently if and only if all symmetries of L preserve some isotropic vector. We obtain a complete classification of Borcherds lattices, consisting of 194 lattices. In turn this provides a classification of hyperbolic lattices with virtually abelian symmetry group and rank ≥5. Finally, we apply these general results to the case of K3 surfaces. We obtain a classification of Picard lattices of K3 surfaces of zero entropy and infinite automorphism group, consisting of 193 lattices. In particular we show that all Kummer surfaces, all supersingular K3 surfaces and all K3 surfaces covering an Enriques surface (with one exception) admit an automorphism of positive entropy.

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