Isoresidual fibration and resonance arrangements
演讲者
时间
2025年04月22日 15:00 至 16:00
地点
A7-101
线上
Zoom 361 038 6975
(BIMSA)
摘要
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.
演讲者介绍
Tahar是BIMSA助理研究员。在加入BIMSA之前，他曾在魏茨曼科学研究所担任高级博士后研究员。他致力于平面上各种几何结构的模空间研究，包括平移和扩张结构、平坦度规和锥球度规。Guillaume-Tahar最近的研究兴趣涉及线性微分算子、isoresidual fibrations和simplicial arrangements of lines.