Isoresidual fibration and resonance arrangements
Organizers
Speaker
Time
Tuesday, April 22, 2025 3:00 PM - 4:00 PM
Venue
A7-101
Online
Zoom 361 038 6975
(BIMSA)
Abstract
Meromorphic 1-forms on the Riemann sphere, with prescribed orders of singularities, form strata equipped with period coordinates. Fixing the residues at the poles defines the isoresidual fibration of any given stratum onto the vector space of residue configurations. Remarkably, the singular locus of this fibration corresponds to an arrangement of complex hyperplanes. In the specific case of 1-forms with only one zero, the fibration becomes a ramified cover. We provide a formula to compute the degree of this cover and analyze its monodromy. The results leverage the correspondence between complex analysis and the flat geometry of translation surfaces.
The qualitative geometry of these translation surfaces is classified via decorated trees, reducing the computation of the cover's degree to a combinatorial problem. For strata with two zeroes, isoresidual fibers are complex curves endowed with a canonical translation structure. The singularities of these fibers encode, through their local invariants, the corresponding degenerations of the parametrized objects. The monodromy is described in terms of a Gauss-Manin connection, which possesses rich geometric and combinatorial properties. This work is a collaboration with Dawei Chen, Quentin Gendron, and Miguel Prado.
The qualitative geometry of these translation surfaces is classified via decorated trees, reducing the computation of the cover's degree to a combinatorial problem. For strata with two zeroes, isoresidual fibers are complex curves endowed with a canonical translation structure. The singularities of these fibers encode, through their local invariants, the corresponding degenerations of the parametrized objects. The monodromy is described in terms of a Gauss-Manin connection, which possesses rich geometric and combinatorial properties. This work is a collaboration with Dawei Chen, Quentin Gendron, and Miguel Prado.
Speaker Intro
Guillaume Tahar obtained his Ph.D from Université Paris Diderot, under the supervision of Anton Zorich. He was a senior postdoctoral fellow in Weizmann Institute of Science and joined BIMSA as an assistant professor in 2022. He contributed to the study of moduli spaces of various flavours of geometric structures on surfaces. His results include proving the existence of closed geodesics in dilation surfaces and the complete characterization of configurations of local invariants realized by a differential on a Riemann surface. His recent research interests include linear differential operators, simplicial arrangements of lines and quantum invariants of knots.