Finite orbits for large groups of automorphisms of projective surfaces

TL;DR

This paper investigates the structure of finite orbits under automorphism groups of projective surfaces, revealing conditions under which these orbits are sparse or dense, with implications for height functions and algebraic dynamics.

Contribution

It establishes new criteria for the distribution of finite orbits in automorphism groups over number fields and complex numbers, highlighting the role of parabolic elements and Kummer examples.

Findings

Finite orbits are not Zariski dense when the group contains parabolic elements over a number field.

In certain rigid cases, finite orbits can be dense, known as Kummer examples.

Results extend to automorphism groups over the complex numbers and relate to canonical height functions.

Abstract

We study finite orbits for non-elementary groups of automorphisms of compact projective surfaces. In particular we prove that if the surface and the group are defined over a number field k and the group contains parabolic elements, then the set of finite orbits is not Zariski dense, except in certain very rigid situations, known as Kummer examples. Related results are also established when k=C. An application is given to the description of "canonical vector heights" associated to such automorphism groups.