Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Isoresidual curves

Dawei Chen, Quentin Gendron, Miguel Prado, Guillaume Tahar

TL;DR

This paper develops a comprehensive framework for isoresidual fibrations over genus-zero strata of meromorphic differentials with two zeros on CP^1. It constructs canonical translation structures on generic isoresidual fibers via relative periods, and analyzes pole/zero configurations, residues, and associated discrete invariants, including a period central charge that commutes with monodromy. By employing the multi-scale compactification, it computes Euler characteristics, describes wall-and-chamber decompositions, and relates these to singularity patterns of the strata; it also provides a Gauss–Manin formalism for monodromy and gives explicit classifications of connected components in genus zero. The results yield precise descriptions of the boundary behavior, pole orders, and pole-residue relations across degenerations, culminating in a complete classification of connected components of one-dimensional generic isoresidual fibers and a detailed account of the orders of singularities in both strata and fibers. Overall, the work bridges flat-geometry, algebraic geometry, and topological invariants to advance understanding of isoresidual fibrations in low-genus differential moduli.

Abstract

Given a partition $μ$ of $-2$, the stratum $\mathcal{H}(μ)$ parametrizes meromorphic differential one-forms on the Riemann sphere $\mathbb{CP}^{1}$ with~$n$ zeros and $p$ poles of orders prescribed by $μ$. The isoresidual fibration is defined by assigning to each differential in $\mathcal{H}(μ)$ its configuration of residues at the poles. In the case of differentials with $n=2$ zeros, generic isoresidual fibers are complex curves endowed with a canonical translation structure, which we describe extensively in this paper. Quantitative characteristics of the translation structure on isoresidual fiber curves, including the orders of the singularities and a period central charge encapsulating the linear dependence of periods on the underlying configuration of residues, provide rich discrete invariants for these fibers. We also determine the Euler characteristic of generic isoresidual fiber curves from intersection-theoretic computations, relying on the multi-scale compactification of strata of differentials. In particular, we describe a wall and chamber structure for the Euler characteristic of generic isoresidual fiber curves in terms of the partition $μ$. Additionally, we classify the connected components of generic isoresidual fibers for strata in genus zero with an arbitrary number of zeros.

Isoresidual curves

TL;DR

This paper develops a comprehensive framework for isoresidual fibrations over genus-zero strata of meromorphic differentials with two zeros on CP^1. It constructs canonical translation structures on generic isoresidual fibers via relative periods, and analyzes pole/zero configurations, residues, and associated discrete invariants, including a period central charge that commutes with monodromy. By employing the multi-scale compactification, it computes Euler characteristics, describes wall-and-chamber decompositions, and relates these to singularity patterns of the strata; it also provides a Gauss–Manin formalism for monodromy and gives explicit classifications of connected components in genus zero. The results yield precise descriptions of the boundary behavior, pole orders, and pole-residue relations across degenerations, culminating in a complete classification of connected components of one-dimensional generic isoresidual fibers and a detailed account of the orders of singularities in both strata and fibers. Overall, the work bridges flat-geometry, algebraic geometry, and topological invariants to advance understanding of isoresidual fibrations in low-genus differential moduli.

Abstract

Given a partition of , the stratum parametrizes meromorphic differential one-forms on the Riemann sphere with~ zeros and poles of orders prescribed by . The isoresidual fibration is defined by assigning to each differential in its configuration of residues at the poles. In the case of differentials with zeros, generic isoresidual fibers are complex curves endowed with a canonical translation structure, which we describe extensively in this paper. Quantitative characteristics of the translation structure on isoresidual fiber curves, including the orders of the singularities and a period central charge encapsulating the linear dependence of periods on the underlying configuration of residues, provide rich discrete invariants for these fibers. We also determine the Euler characteristic of generic isoresidual fiber curves from intersection-theoretic computations, relying on the multi-scale compactification of strata of differentials. In particular, we describe a wall and chamber structure for the Euler characteristic of generic isoresidual fiber curves in terms of the partition . Additionally, we classify the connected components of generic isoresidual fibers for strata in genus zero with an arbitrary number of zeros.

Paper Structure

This paper contains 59 sections, 50 theorems, 71 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of the paper
  4. Acknowledgements.
  5. Meromorphic differentials and translation structures
  6. Translation structures
  7. The regular locus
  8. Local models for singularities
  9. Topological index of a loop in a translation surface
  10. Period coordinates for strata
  11. The period atlas
  12. Saddle connections
  13. Cylinders
  14. The action of ${\rm GL}_{2}^{+}(\mathbb{R})$
  15. Core
  16. ...and 44 more sections

Key Result

Theorem 1.2

For a stratum $\mathcal{H}(a_{1},a_{2},-b_{1},\dots,-b_{p})$ and a configuration $\lambda \in \mathcal{R}_{p} \setminus \mathcal{A}_{p}$, the closure $\overline{\mathcal{F}}_{\lambda}$ of the isoresidual fiber $\mathcal{F}_{\lambda}$ is a (possibly disconnected) compact Riemann surface endowed with

Figures (10)

  • Figure 1: A differential $\omega$ and the associated ribbon and decorated graphs are shown, with extra labels on the edges given by the label of the corresponding saddle connection.
  • Figure 2: A differential in $\mathcal{H}(1,4,-1,-2,-4)$ and its associated graphs are presented.
  • Figure 3: The level graphs $\Gamma$ and $\Gamma'$, illustrating the $\mathcal{E}_{\Lambda}$-GRC. The vertices at level infinity are pictured in white.
  • Figure 4: The types of level graphs in Proposition \ref{['prop:F-boundary']}.
  • Figure 5: A differential in $\mathcal{H}(2,3,-1,-2,-4)$ such that the isoresidual deformations are given by varying $\gamma$ only.
  • ...and 5 more figures

Theorems & Definitions (125)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 115 more