Let X be a normal quasi-projective variety over C. We study its higher Albanese manifolds, introduced by Hain and Zucker, from the point of view of o-minimal geometry. We show that for each s the higher Albanese manifold Albˢ (X) can be functorially endowed with a structure of an R₀₋₆-definable complex manifold in such a way that the natural projections Albˢ (X) Alb^s-1 (X) are R₀₋₆-definable and the higher Albanese maps albˢ X^an Albˢ (X) are R₀₍, ₄ₗ-definable. Suppose that for some s 3 the definable manifold Albˢ (X) is definably biholomorphic to a quasi-projective variety. We show that in this case the higher Albanese tower stabilises at the second step, i. e. the maps Albʳ (X) Alb^r-1 (X) are isomorphisms for r 3. It follows that if albˢ X^an Albˢ (X) is dominant for some s 3, then the higher Albanese tower stabilises at the second step and the pro-unipotent completion of π₁ (X) is at most 2-step nilpotent. This confirms a special case of a conjecture by Campana on nilpotent fundamental groups of algebraic varieties. As another application, we prove the existence and quasi-projectivity of unipotent Shafarevich reductions.
Vasily Rogov (Mon,) studied this question.