The nilpotent quotients of normal quasi-projective varieties with proper quasi-Albanese map

Abstract

We show that if X is a smooth complex quasi-projective manifold the quasi-Albanese map of which is proper, then the torsionfree nilpotent quotients of π 1 (X) are the same ones as those of the Stein factorisation of its quasi-Albanese image. When X is quasi-Kähler smooth, we get the same conclusion, but only for the smooth models of the Stein factorisation. The proof is then completely similar to the completely elementary one given in [6] for X compact.This implies in the quasi-projective case that the étale Galois cover of X associated to the nilpotent completion of π 1 (X) is holomorphically convex. This last result is proved in the quasiprojective case by 3 other methods in [14], which motivated the present text.When X is 'special' in the sense of [8], we deduce that the torsion free nilpotent quotients of π 1 (X) are abelian, and show by examples that this property fails (as first observed in [5]) when the quasi-Albanese map is not proper. This leads to revise our previous 'Abelianity conjecture', and replace it by an 'Nilpotency conjecture' in the quasi-Kähler context.