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

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.