The nilpotent quotients of normal quasi-projective varieties with proper quasi-Albanese map
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 1 (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-Khler 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 [8] for X compact.In the normal quasi-projective case, the tale Galois cover of X associated to the nilpotent completion of 1 (X) is thus holomorphically convex.This is proved in the smooth case by 3 other methods in [18], which motivated the present text.When X is 'special' in the sense of [11], we deduce that the torsion free nilpotent quotients of 1 (X) are abelian.Examples show that this property fails (as first observed in [6]) 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-Khler context. 1 Introduction 912 2 Proof of Theorem 2 914 3 Proof of Theorem 4 916 4 Proof of Theorem 6 920 5 Special manifolds 921
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