Isoresidual fibration and resonance arrangements

TL;DR

This paper investigates the structure of the isoresidual fibration on certain meromorphic differential strata on the Riemann sphere, revealing its unramified cover properties and monodromy through combinatorial and geometric analysis.

Contribution

It establishes that the isoresidual fibration is an unramified cover of a specific degree outside a resonance hyperplane arrangement and computes its monodromy for strata with up to three poles.

Findings

The isoresidual fibration is an unramified cover of degree a!/(a+2-p)!

Monodromy is explicitly computed for strata with up to three poles

A combinatorial model using trees relates geometric properties to arrangements

Abstract

The stratum $\mathcal{H}(a,-b_{1},\dots,-b_{p})$ of meromorphic $1$-forms with a zero of order $a$ and poles of orders $b_{1},\dots,b_{p}$ on the Riemann sphere has a map, the isoresidual fibration, defined by assigning to any differential its residues at the poles. We show that above the complement of a hyperplane arrangement, the resonance arrangement, the isoresidual fibration is an unramified cover of degree $\frac{a!}{(a+2-p)!}$. Moreover, the monodromy of the fibration is computed for strata with at most three poles and a system of generators and relations is given for all strata. These results are obtained by associating to special differentials of the strata a tree, and by studying the relationship between the geometric properties of the differentials and the combinatorial properties of these trees.