Parabolic automorphisms of hyperk{ä}hler manifolds: Orbits and Betti maps

TL;DR

The paper studies parabolic automorphisms of irreducible hyperkähler manifolds with lagrangian fibrations, focusing on their action as fiberwise translations on smooth fibers and the associated Betti maps. It introduces Betti coordinates and a translation vector $t_f$, proving that $t_f$ has maximal variation (rank $2g$) in the projective and non-projective settings, which yields density results for fibers where the induced translation is torsion. A novel, self-contained approach is developed that combines volume estimates, polarized endomorphisms, and cohomological growth to control the Betti map without relying on Ax–Schanuel-type theorems, and it extends to non-projective (Kähler) manifolds via degenerate twistor deformations. The results unify the projective and non-projective theories, extend Lo Bianco’s theorem to the non-projective case, and provide concrete dynamical consequences such as dense orbit closures and the density of fibers with finite-order translations.

Abstract

We study parabolic automorphisms of irreducible holomorphically symplectic manifolds with a lagrangian fibration. Such automorphisms are (possibly up to taking a power) fiberwise translations on smooth fibers, and their orbits in a general fiber are dense ([1]). We provide a simple proof that the associated Betti map is of maximal rank, in particular, the set of fibers where the induced translation is of finite order is dense as well. R{É}SUM{É}. Nous {é}tudions les automorphismes paraboliques des vari{é}t{é}s symplectiques holomorphes qui sont irr{é}ductibles et projectives.

Theorems & Definitions (19)

• Example 1.1
• Remark 1.2
• proof
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Proposition 4.1
• Lemma 4.2
• Proposition 4.3