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 $\nu :{\pi }_1(X)\twoheadrightarrow H$, where $H$ is a finitely generated abelian group with $\mathrm{rank}H\ge 1$. In this paper, we study the asymptotic behaviour of Betti numbers with all possible field coefficients and the order of the torsion subgroup of singular homology associated to $\nu$, known as the $L^2$-type invariants. When $\nu$ is orbifold effective, we give explicit formulas of these invariants at degree 1. This generalizes the authors' previous work for $H\cong \mathbb{Z}$.

12 pages, to appear in Chinese Annals of Mathematics, series B. arXiv admin note: text overlap with arXiv:2110.03356