Vasily Rogov

Classical Albanese manifold together with the Albanese map can be viewed as a tool that geometrically encodes the Hodge structure on first homology of a compact Kähler manifold or  a normal  complex quasi-projective variety. Hain and Zucker constructed a tower of complex manifolds, called higher Albanese manifolds, that play similar role for mixed Hodge structure on nilpotent quotients of the fundamental group. In contrast to the classical situation, their construction is a priori merely complex analytic and it turns out, that higher Albanese manifolds are rarely algebraic varieties. I will explain, that this theory nevertheless fits into the world of o-minimal complex analytic geometry, which gives control on behaviour of higher Albanese maps at infinity. As a consequence, we will see, that if a higher Albanese map $alb^s$ is dominant for some s greater than 2, then the higher Albanese tower stabilises at step 2 and every torsion free nilpotent quotient of $\pi_1(X)$ is at most 2 step nilpotent. This gives new insight towards the Nilpotency Conjecture of Campana.