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

Let X be a complex smooth quasi-projective variety with an epimorphism ν π₁(X) 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.