Actions of nilpotent groups on complex algebraic varieties

Marc Abboud (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We study nilpotent groups acting faithfully on complex algebraic varieties. We use a method of base change. For finite p-groups, we go from k, a number field, to a finite field in order to use counting lemmas. We show that a finite p-group of polynomial automorphisms of k^d is isomorphic to a subgroup of GL d (k). For infinite groups, we go from C to Z_p and use p-adic analytic tools and the theory of p-adic Lie groups. We show that a finitely generated nilpotent group H acting faithfully on a complex quasiprojective variety X of dimension d can be embedded into a p-adic Lie group acting faithfully and analytically on Z_p^d ; we deduce that d is larger than the virtual derived length of H.

Mots clés: Algebraic geometry; Arithmetic and non-Archimedean dynamical systems; Group actions on varieties or schemes; Arithmetic dynamics on general algebraic varieties

Domaines: Géométrie algébrique [math.AG]; Systèmes dynamiques [math.DS]

Marc Abboud. Actions of nilpotent groups on complex algebraic varieties. International Mathematics Research Notices, 2022, ⟨10.1093/imrn/rnac056⟩. ⟨hal-03543848⟩