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

TL;DR

This work develops a comprehensive framework for $L^2$-type invariants of complex smooth quasi-projective varieties via a fixed epimorphism $\nu: \pi_1(X) \twoheadrightarrow \mathbb{Z}$. It connects asymptotic Betti numbers and torsion growth to cohomology jump loci and Alexander polynomials, and derives explicit degree-one formulas when $\nu$ is orbifold effective, expressed in terms of orbifold data $(g,r,\boldsymbol\mu)$. A core contribution is the generalization of Arapura-type results to arbitrary algebraically closed fields and the demonstration that, under suitable hypotheses, the degree-one invariants are controlled by orbifold maps, including hyperbolic maps, with concrete bounds and cyclotomic-type Alexander polynomials. The paper then applies these ideas to hyperplane arrangement complements, establishing combinatorial upper bounds for parallel components of $\mathcal{V}^1(X,\mathbb{C})$ and answering a question of Denham and Suciu about Milnor fiber torsion in certain central arrangements. The results unify $L^2$-invariants, Aomoto theory, and orbifold geometry to produce both general theorems and explicit invariants in concrete geometric settings.

Abstract

Let X be a complex smooth quasi-projective variety with a fixed epimorphism $ν\colonπ_1(X)\twoheadrightarrow \mathbb{Z}$. 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 $ν$, 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 $ν$ 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.

Theorems & Definitions (62)

• Proposition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Definition 1.8
• Remark 1.9
• Theorem 1.10