Isoresidual fibration and resonance arrangements

Abstract

Meromorphic 1-forms on the Riemann sphere with prescribed orders of singularities form strata endowed with period coordinates. Fixing residues at the poles defined a fibration of any stratum to the vector space of configurations of residues. For strata of 1-forms with only one zero, the isoresidual fibration is a cover of the space of configurations of residues ramified over an arrangement of complex hyperplanes. We give a formula to compute the degree of this cover and investigate its monodromy.The results are obtained using the dictionary between complex analysis and flat geometry of translation surfaces. The qualitative geometry of the latter translation surfaces is then classified by decorated trees, reducing the computation of the degree of the cover to a combinatorial problem. For strata with several zeroes, isoresidual fibers are complex manifolds endowed with a matrix-valued meromorphic differential. Singularities of the differential give insights on the topological invariants of the fibers. This is a joint work with Quentin Gendron.