$L^2$-type invariants and cohomology jump loci for complex smooth quasi-projective varieties

TL;DR

This paper investigates the asymptotic behavior of $L^2$-type invariants and cohomology jump loci for complex smooth quasi-projective varieties, providing formulas, extending results, and applying findings to hyperplane arrangements.

Contribution

It introduces new formulas for $L^2$-type invariants in degree one, extends Arapura's results to positive characteristic, and applies these to hyperplane arrangements.

Findings

Concrete formulas for asymptotic invariants in degree one.

Extension of cohomology jump loci results to positive characteristic.

A combinatorial upper bound for hyperplane arrangements.

Abstract

Let X be a complex smooth quasi-projective variety with a fixed epimorphism $\nu\colon\pi_1(X)\twoheadrightarrow \mathbb{Z}$. In this paper, we consider the asymptotic behaviour of invariants such as Betti numbers with all possible field coefficients and the order of the torsion part of singular integral homology associated to $\nu$, known as the $L^2$-type invariants. At homological degree one, we give concrete formulas for these limits by the geometric information of $X$ when $\nu$ is orbifold effective. The proof relies on a study about cohomological degree one jump loci of $X$. We extend part of Arapura's result for cohomological degree one jump loci of $X$ with complex field coefficients to the one with positive characteristic field coefficients. As an application, when $X$ 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.