2023/01/26 by Rodolfo Aguilar, Frédéric Campana, Aguilar, Rodolfo Aguilar +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Geometry and complex manifolds #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2301.11232
We show that if X is a normal complex quasi-projective variety, the quasi-Albanese map of which is proper, then the torsionfree nilpotent quotients of π1(X) are, up to a controlled finite index, the same ones as those of the normalisation of its quasi-Albanese image. When X is quasi-Kähler smooth, we get the same conclusion, but only for the smooth models of the quasi-Albanese image. In this second case, the proof is elementary, as the one given in [Cam95] for X compact. In the normal quasi-projective case, the étale Galois cover of X associated to the nilpotent completion of π1(X) is thus holomorphically convex. This is proved in the smooth case by 3 other methods in [GGK22], which motivated the present text. When X is `special' in the sense of [Cam11], we deduce that the torsion free nilpotent quotients of π1(X) are abelian. Examples show that this property fails (as first observed in [CDY22]) when the quasi-Albanese map is not proper. This leads to replace our previous `Abelianity conjecture' in the compact case by an `Nilpotency conjecture' in the non-compact quasi-Kähler context.