O-minimal geometry of higher Albanese manifolds

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 Ralg-definable complex manifold in such a way that the natural projections Albˢ(X) → Albˢ⁻¹(X) are Ralg-definable and the higher Albanese maps albˢ Xᵃⁿ → Albˢ(X) are Ran, exp-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) → Albʳ⁻¹(X) are isomorphisms for r≥ 3. It follows that if albˢ Xᵃⁿ → 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.