L²-type invariants for complex smooth quasi-projective varieties -- a survey

2024/06/18 by Yongqiang Liu, Liu, Yongqiang
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2406.12287

openalex publication_date 2024/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a complex smooth quasi-projective variety with an epimorphism ν\colon π₁(X)\twoheadrightarrow ℤⁿ. 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 ν, known as the L²-type invariants. We give relations between L²-type invariants, Alexander invariants and cohomology jump loci. When ν is orbifold effective, we give explicit formulas for L²-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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