O-minimal geometry of higher Albanese manifolds

Let X be a normal quasi-projective variety over $\mathbb{C}$. 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 $\operatorname{Alb}^s(X)$ can be functorially endowed with a structure of an $\mathbb{R}_{\operatorname{alg}}$-definable complex manifold in such a way that the natural projections $\operatorname{Alb}^s(X) \to \operatorname{Alb}^{s-1}(X)$ are $\mathbb{R}_{\operatorname{alg}}$-definable and the higher Albanese maps $\operatorname{alb}^s \colon X^{\operatorname{an}} \to \operatorname{Alb}^s(X)$ are $\mathbb{R}_{\operatorname{an}, \operatorname{exp}}$-definable. Suppose that for some $s \ge 3$ the definable manifold $\operatorname{Alb}^s(X)$ 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 $\operatorname{Alb}^r (X) \to \operatorname{Alb}^{r-1}(X)$ are isomorphisms for $r\ge 3$. It follows that if $\operatorname{alb}^s \colon X^{\operatorname{an}} \to \operatorname{Alb}^s(X)$ is dominant for some $s \ge 3$, then the higher Albanese tower stabilises at the second step and the pro-unipotent completion of $\pi_1(X)$ 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.