Finite orbits for large groups of automorphisms of projective surfaces

Serge Cantat (1) , Romain Dujardin (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

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.

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]; Systèmes dynamiques [math.DS]

Serge Cantat, Romain Dujardin. Finite orbits for large groups of automorphisms of projective surfaces. Compositio Mathematica, 2023, 160 (1), pp.120-175. ⟨10.1112/S0010437X23007613⟩. ⟨hal-03035398⟩