Isoresidual fibration and resonance 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.

arXiv commentFinal version. To appear in: Letters in Mathematical Physics