On the Kobayashi pseudometric, complex automorphisms and hyperkaehler manifolds

TL;DR

This paper explores the properties of the Kobayashi pseudometric and quotient in complex manifolds, especially hyperkähler manifolds, revealing new insights into their automorphisms, fibrations, and dynamical behavior.

Contribution

It introduces the Kobayashi quotient for complex varieties, proves isomorphism of quotients for ergodic structures, and confirms Kobayashi's conjecture for hyperkähler manifolds with specific fibrations.

Findings

Kobayashi quotients are nontrivial for manifolds with infinite order automorphisms.

Kobayashi quotients for ergodic structures are isomorphic.

Hyperkähler manifolds with certain fibrations have vanishing pseudodistance.

Abstract

We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them and show that if a compact complex manifold has an automorphism whose order is infinite, then the fibers of this quotient map are nontrivial. We prove that the Kobayashi quotients associated to ergodic complex structures on a compact manifold are isomorphic. We also give a proof of Kobayashi's conjecture on the vanishing of the pseudodistance for hyperk\"ahler manifolds having Lagrangian fibrations without multiple fibers in codimension one. For a hyperbolic automorphism of a hyperk\"ahler manifold, we prove that its cohomology eigenvalues are determined by its Hodge numbers, compute its dynamical degree and show that its cohomological trace grows exponentially, giving estimates on the number of its periodic points.