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

TL;DR

This paper investigates the structure of the fundamental group of certain complex algebraic varieties with proper quasi-Albanese maps, showing that their nilpotent quotients are closely related to the quasi-Albanese image, with implications for holomorphic convexity and group properties.

Contribution

It establishes a connection between the nilpotent quotients of the fundamental group and the quasi-Albanese image for varieties with proper quasi-Albanese maps, extending previous results and proposing a new conjecture.

Findings

Nilpotent quotients of $\pi_1(X)$ match those of the quasi-Albanese image up to finite index.

The associated Galois cover of $X$ is holomorphically convex.

For 'special' varieties, the nilpotent quotients are abelian.

Abstract

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 $\pi_1(X)$ are, up to a controlled finite index, the same ones as those of the normalisation of its quasi-Albanese image. When $X$ is quasi-K\"ahler 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 \'etale Galois cover of $X$ associated to the nilpotent completion of $\pi_1(X)$ 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 $\pi_1(X)$ 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-K\"ahler context.