Parabolic automorphisms of hyperk{\"a}hler manifolds: Orbits and Betti maps

TL;DR

This paper investigates parabolic automorphisms of hyperk{"a}hler manifolds with Lagrangian fibrations, showing that their associated Betti maps are of maximal rank and that fibers with finite order translations are dense.

Contribution

It provides a simple proof that the Betti map associated with such automorphisms has maximal rank and demonstrates the density of fibers with finite order translations.

Findings

Betti map is of maximal rank for these automorphisms

Fibers with finite order translations are dense

Automorphisms act as fiberwise translations on smooth fibers

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{\'E}SUM{\'E}. Nous {\'e}tudions les automorphismes paraboliques des vari{\'e}t{\'e}s symplectiques holomorphes qui sont irr{\'e}ductibles et projectives.