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Non-projective K3 surfaces with real or Salem multiplication

Eva Bayer-Fluckiger, Bert van Geemen, Matthias Schütt

TL;DR

This work extends Zarhin’s RM/CM framework to non-projective K3 surfaces and hyperkähler manifolds by introducing pseudo-polarized Hodge structures and a transfer mechanism from quadratic/hermitian forms over endomorphism fields. It proves a dichotomy: the Hodge endomorphism algebra A_X is either a totally real field or a Salem field, constraining the transcendental lattice via the dimension m=dim_E(T_Q). The authors establish existence results for realizations of RM and SM across K3 surfaces and HK-manifolds, derive precise criteria involving determinants, discriminants, and local invariants, and connect these arithmetic structures to dynamics and number theory (e.g., Salem numbers, Stark-Chinburg theory). Collectively, the paper links Hodge theory, lattice theory, and dynamics to classify endomorphism algebras in the non-projective setting and to illuminate rich interactions with Salem fields and dynamical degrees of automorphisms.

Abstract

We determine the Hodge endomorphism algebras of non-projective complex K3 surfaces (and more generally, hyperkähler manifolds). We show that they are either totally real fields or number fields generated by Salem numbers. This is unlike the projective case, where the endomorphism fields are either totally real or CM. We also develop precise existence criteria and explore the relations to number theory and dynamics.

Non-projective K3 surfaces with real or Salem multiplication

TL;DR

This work extends Zarhin’s RM/CM framework to non-projective K3 surfaces and hyperkähler manifolds by introducing pseudo-polarized Hodge structures and a transfer mechanism from quadratic/hermitian forms over endomorphism fields. It proves a dichotomy: the Hodge endomorphism algebra A_X is either a totally real field or a Salem field, constraining the transcendental lattice via the dimension m=dim_E(T_Q). The authors establish existence results for realizations of RM and SM across K3 surfaces and HK-manifolds, derive precise criteria involving determinants, discriminants, and local invariants, and connect these arithmetic structures to dynamics and number theory (e.g., Salem numbers, Stark-Chinburg theory). Collectively, the paper links Hodge theory, lattice theory, and dynamics to classify endomorphism algebras in the non-projective setting and to illuminate rich interactions with Salem fields and dynamical degrees of automorphisms.

Abstract

We determine the Hodge endomorphism algebras of non-projective complex K3 surfaces (and more generally, hyperkähler manifolds). We show that they are either totally real fields or number fields generated by Salem numbers. This is unlike the projective case, where the endomorphism fields are either totally real or CM. We also develop precise existence criteria and explore the relations to number theory and dynamics.

Paper Structure

This paper contains 46 sections, 63 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Hodge structures of K3 type
  3. K3 surfaces
  4. The sublattices $\operatorname{Pic}(X)$ and $T_X$
  5. The Hodge endomorphisms
  6. K3 surfaces of algebraic dimension 1
  7. Pseudo-polarized Hodge structures
  8. K3 surfaces of algebraic dimension 0
  9. Quadratic and hermitian forms, transfer
  10. Pseudo-polarized Hodge structures and hermitian forms
  11. Building blocks for signatures
  12. Endomorphism fields of pseudo-polarized Hodge structures
  13. Proof of Theorem \ref{['endos']}
  14. From quadratic forms to pseudo-polarized Hodge structures
  15. Signatures
  16. ...and 31 more sections

Key Result

Theorem 2

Let $X$ be a K3 surface with $a(X) = 0$, or more generally a hyperkähler manifold $X$ with pseudo-polarized Hodge structure $T_{X,{\bf{Q}}}$. Let $E =A_X$ be the algebra of Hodge endomorphisms of $T_{X,{\bf{Q}}}$, and set $m = {\rm dim}_E(T_{X,{\bf{Q}}})$. Then $E$ is a number field with a $\bf Q$-l

Theorems & Definitions (82)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Proposition 6
  • Theorem 7
  • Proposition 1.1
  • Lemma 1.2
  • Remark 1.3
  • ...and 72 more