Profinite Non-Rigidity of Arithmetic Lattices and the {K}\"ahler Property

We prove that the K"ahlerness of a finitely presented, residually finite group is not determined by its profinite completion, which answers a question asked independently by Arapura and Libgober. In the proof, we give explicit constructions to characterize the absolute Dynkin types that admit non-isomorphic real forms of higher-rank Lie groups containing torsion-free arithmetic lattices with isomorphic profinite completions.