$L^2$-type invariants for complex smooth quasi-projective varieties -- a survey

Yongqiang Liu

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arXiv · 2406.12287

$L^2$-type invariants for complex smooth quasi-projective varieties -- a survey

Yongqiang Liu

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Abstract

Let X be a complex smooth quasi-projective variety with an epimorphism $\nu \colon \pi_1(X)\twoheadrightarrow \mathbb{Z}^n$. We survey recent developments about the asymptotic behaviour of Betti numbers with any field coefficients and the order of the torsion part of singular integral homology of finite abelian covers of $X$ associated to $\nu$, known as the $L^2$-type invariants. We give relations between $L^2$-type invariants, Alexander invariants and cohomology jump loci. When $\nu$ is orbifold effective, we give explicit formulas for $L^2$-invariants at homological degree one in terms of geometric information of $X$. We also propose several related open questions for hyperplane arrangement complement.

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Yongqiang Liu. 2024-06-18. $L^2$-type invariants for complex smooth quasi-projective varieties -- a survey. https://arxiv.org/abs/2406.12287

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