Characteristic varieties and logarithmic differential 1-forms

TL;DR

This paper explores the relationship between characteristic and resonance varieties of local systems on complex varieties, introducing a hypercohomology approach and applying it to configuration spaces on elliptic curves.

Contribution

It develops a hypercohomology framework for resonance varieties, linking them to characteristic varieties, and applies this to specific geometric configurations and forms.

Findings

Determined the first characteristic variety of configuration spaces on elliptic curves.

Established relations between resonance degree and zero set codimension of logarithmic forms.

Introduced a hypercohomology version of resonance varieties.

Abstract

We introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety $M$, see Theorem (3.1) and Corollaries (3.2) and (4.2). A logarithmic resonance variety is also considered in Proposition (4.5). As an application, we determine the first characteristic variety of the configuration space of $n$ distinct labeled points on an elliptic curve, see Proposition (5.1). Finally, for a logarithmic one form $\alpha$ on $M$ we investigate the relation between the resonance degree of $\alpha$ and the codimension of the zero set of $\alpha$ on a good compactification of $M$, see Corollary (1.1). This question was inspired by the recent work by D. Cohen, G. Denham, M. Falk and A. Varchenko.