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

Abstract

Let $X$ be a normal complex algebraic variety. Let $\mathcal{G}^s_{\mathbb{Z}}(X)$ be the maximal torsion free nilpotent quotient of $π_1(X)$ 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 $π_1(X)$. We show that the natural map $H^k(\mathcal{G}^s_{\mathbb{Z}}(X), \mathbb{Z}) \to H^k(X, \mathbb{Z})$ vanishes for $k > \dim F^1\mathfrak{g}^s$. If $\mathcal{G}^s_{\mathbb{Z}}(X)$ is of nilpotency class greater than two, this includes the top nonvanishing degree of $H^{\bullet}(\mathcal{G}^s_{\mathbb{Z}}(X), \mathbb{Z})$. 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.

Disclosure

“gs ´dim F 0 gs . Using Hamm’s theorem on the homotopy type of q-complete spaces, we then bound the cohomological dimension of Y . Acknowledgements. I am thankful to MPI MiS, Leipzig, for great working conditions. AI Usage Disclosure. Claude Sonnet 5 (Anthropic) was used to improve style, grammar, and organization, and to assist in developing several motivating examples; the author takes full responsibility for the content of the paper. 2. Higher Albanese”

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2026-08-23T19:02:23.172603Z

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