L2-type invariants and cohomology jump loci for complex smooth quasi-projective varieties

Let [Formula: see text] be a complex smooth quasi-projective variety with a fixed epimorphism [Formula: see text]. We consider the asymptotic behavior of invariants such as Betti numbers with any field coefficients and the order of the torsion part of singular integral homology associated to [Formula: see text], known as the [Formula: see text]-type invariants. At homological degree one, we give concrete formulas for these limits by the geometric information of [Formula: see text] when [Formula: see text] is orbifold effective. The proof relies on a study about cohomological degree one jump loci of [Formula: see text]. We extend part of Arapura’s result for cohomological degree one jump loci of [Formula: see text] with complex field coefficients to the one with positive characteristic field coefficients. As an application, when [Formula: see text] is a hyperplane arrangement complement, a combinatoric upper bound is given for the number of parallel positive-dimensional components in cohomological degree one jump loci with complex coefficients. Another application is that we give a positive answer to a question posed by Denham and Suciu for hyperplane arrangement.