Profinite rigidity of K\"ahler groups: Riemann surfaces and subdirect products

TL;DR

This paper proves that certain K"ahler groups, including products of surface groups, are uniquely determined by their profinite completions, leading to new insights into the algebraic and topological properties of associated varieties.

Contribution

It establishes strong profinite rigidity results for K"ahler groups, showing they are determined by their profinite completions, and recovers holomorphic fibrations from these completions.

Findings

Profinite completions determine holomorphic fibrations over hyperbolic 2-orbifolds.

Certain K"ahler groups are uniquely identified by their profinite completions.

Profinite invariance of the BNS invariant is proven.

Abstract

This paper establishes strong profinite rigidity results for K\"ahler groups, showing that certain groups are determined within the class of residually finite K\"ahler groups by their profinite completion. Examples include products of surface groups and certain groups with exotic finiteness properties studied earlier by Dimca-Papadima-Suciu and Llosa Isenrich. Consequently, there are aspherical smooth projective varieties that are determined up to homeomorphism by their algebraic fundamental group. The main tool is the following: the holomorphic fibrations of a closed K\"ahler manifold over hyperbolic 2-orbifolds can be recovered from the profinite completion of its fundamental group. We also prove profinite invariance of the BNS invariant.