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

2026/07/21 by Vasily Rogov
#math.AG #math.CV

paper · pdf

Abstract

Let X be a normal complex algebraic variety. Let G^s_ℤ(X) be the maximal torsion free nilpotent quotient of π₁(X) of nilpotency class at most s. Let F^\bullet\mathfrakg^s be the Morgan--Hain Hodge filtration on the Lie algebra of the s-th lower central quotient of the complex Malcev completion of π₁(X). We show that the natural map H^k(G^s_ℤ(X), ℤ) → H^k(X, ℤ) vanishes for k > dim F¹\mathfrakg^s. If G^s_ℤ(X) is of nilpotency class greater than two, this includes the top nonvanishing degree of H^\bullet(G^s_ℤ(X), 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.

Citations

• An explicit class of Lagrangian surfaces

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