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<p>Let <i>X</i> be a complex smooth quasi-projective variety with a fixed epimorphism <i>ν</i>: <i>π</i><sub>1</sub>(<i>X</i>) ↠ <i>H</i>, where <i>H</i> is a finitely generated abelian group with rank <i>H</i> ≥ 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 <i>ν</i>, known as the <i>L</i><sup>2</sup>-type invariants. When <i>ν</i> is orbifold effective, explicit formulas of these invariants at degree 1 are give. This generalizes the authors’ previous work for <i>H</i> ≌ ℤ.</p>

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

  • 期刊论文

  • 发表日期：

    2025年1月1日

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  • Fenglin Li,
  • Yongqiang Liu

摘要

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.