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Gaps in the support of canonical currents on projective K3 surfaces

Simion Filip, Valentino Tosatti

Abstract

We construct examples of canonical closed positive currents on projective K3 surfaces that are not fully supported on the complex points. The currents are the unique positive representatives in their cohomology classes and have vanishing self-intersection. The only previously known such examples were due to McMullen on non-projective K3 surfaces and were constructed using positive entropy automorphisms with a Siegel disk. Our construction is based on a Zassenhaus-type estimate for commutators of automorphisms.

Gaps in the support of canonical currents on projective K3 surfaces

Abstract

We construct examples of canonical closed positive currents on projective K3 surfaces that are not fully supported on the complex points. The currents are the unique positive representatives in their cohomology classes and have vanishing self-intersection. The only previously known such examples were due to McMullen on non-projective K3 surfaces and were constructed using positive entropy automorphisms with a Siegel disk. Our construction is based on a Zassenhaus-type estimate for commutators of automorphisms.
Paper Structure (21 sections, 9 theorems, 25 equations)

This paper contains 21 sections, 9 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Gaps in the support of canonical currents
  3. Outline
  4. Commutator estimates
  5. Derived series and commutators
  6. Pseudogroup of transformations
  7. (2,2,2)-surfaces and canonical currents
  8. Setup
  9. Some recollections from topology
  10. Involutions
  11. Canonical currents
  12. Free subgroups of automorphisms
  13. Largeness of the set currents with gaps
  14. An example with slow commutators
  15. Setup
  16. ...and 6 more sections

Key Result

Theorem 1

There exists a projective K3 surface $X$ of type $(2,2,2)$, and an uncountable dense $F_{\sigma}$ set of rays $F\subset \partial \mathop{\mathrm{Amp}}\nolimits(X)$ in the boundary of its ample cone, such that for every $f\in F$ the topological support of the unique canonical current $\eta_f$ is not

Theorems & Definitions (19)

  • Theorem 1: Gaps in the support
  • Theorem 2: Full gaps in the real locus
  • Proposition 1: Fast ramification
  • proof
  • Theorem 2.1.1: Common domain of definition
  • proof
  • Proposition 2: Fixed points with small derivative
  • proof
  • Definition 1: Strict $(2,2,2)$ example
  • Proposition 3: Free group on five generators
  • ...and 9 more