Topology of higher Albanese maps and aspherical varieties with nilpotent fundamental group

Abstract

Let $X$ be a normal complex algebraic variety. Let ${\mathcal{G}}_{\mathbb{Z}}^s(X)$ be the maximal torsion free nilpotent quotient of ${\pi }_1(X)$ of nilpotency class at most $s$. Let $F^{\bullet }{\mathfrak{g}}^s$ be the Morgan--Hain Hodge filtration on the Lie algebra of the $s$-th lower central quotient of the complex Malcev completion of ${\pi }_1(X)$. We show that the natural map $H^k({\mathcal{G}}_{\mathbb{Z}}^s(X),\mathbb{Z})\to H^k(X,\mathbb{Z})$ vanishes for $k>\dim F^1{\mathfrak{g}}^s$. If ${\mathcal{G}}_{\mathbb{Z}}^s(X)$ is of nilpotency class greater than two, this includes the top nonvanishing degree of $H^{\bullet }({\mathcal{G}}_{\mathbb{Z}}^s(X),\mathbb{Z})$. We deduce that if the fundamental group of an aspherical normal variety is virtually nilpotent, it is virtually two-step nilpotent. This gives a positive answer to a question of Aguilar and Campana in this case of aspherical varieties. The ingredients of the proof are the $q$-convexity of higher Albanese manifolds and the definability of higher Albanese maps.

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