The Homology Growth for Finite Abelian Covers of Smooth Quasi-projective Varieties

Abstract

Let X be a complex smooth quasi-projective variety with a fixed epimorphism ν: π₁(X) ↠ H, where H is a finitely generated abelian group with rank H ≥ 1. In this paper, the authors study the asymptotic behaviour of Betti numbers with all possible field coefficients and the order of the torsion subgroup of singular homology associated to ν, known as the L²-type invariants. When ν is orbifold effective, explicit formulas of these invariants at degree 1 are give. This generalizes the authors’ previous work for H ≌ ℤ.

Keywords

Mahler measure / Jump loci / Orbifold map / L²-Betti number / 14F45 / 32S20 / 14H30 / 57Q99

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