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Finite isoresidual covers in strata of $k$-differentials

Dawei Chen, Quentin Gendron, Miguel Prado, Guillaume Tahar

TL;DR

This work studies the isoresidual fibration for strata of primitive $k$-differentials on the sphere with two marked singularities, where residues at poles of order divisible by $k$ are allowed to vary. It introduces a detailed framework—comprising resonance stratification, the multi-scale compactification ${\rm MS}^{k}(\mu)$, and intersection-theoretic counting—to compute the degree of the residue map, including a closed formula $d_k(\mu)=\sum_{c_{1,I}>0} c_{1,I} f_k(a_1,|I|+1) f_k(a_2,|I^c|+1)$ and its resonant refinements via $\operatorname{Ab}_{\mathcal{R}}(I)$. The paper also provides flat-geometric proofs in special cases, derives an alternative gamma-function expression, and analyzes degenerations and resonance to describe ramification along resonance loci. As an application, it counts cone spherical metrics with dihedral monodromy, yielding explicit numbers in terms of the orders of the singularities. Overall, the work extends the abelian ($k=1$) results to higher $k$, linking residue data, intersection theory, and flat-geometry in the study of $k$-differentials.

Abstract

Consider the strata of primitive $k$-differentials on the Riemann sphere whose singularities, except for two, are poles of order divisible by $k$. The map that assigns to each $k$-differential the $k$-residues at these poles is a ramified cover of its image. Generalizing results known in the case of abelian differentials, we describe the ramification locus of this cover and provide a formula, involving the $k$-factorial function, for the cardinality of each fiber. We prove this formula using intersection calculations on the multi-scale compactification of the strata of $k$-differentials. In special cases, we also give alternative proofs using flat geometry. Finally, we present an application to cone spherical metrics with dihedral monodromy.

Finite isoresidual covers in strata of $k$-differentials

TL;DR

This work studies the isoresidual fibration for strata of primitive -differentials on the sphere with two marked singularities, where residues at poles of order divisible by are allowed to vary. It introduces a detailed framework—comprising resonance stratification, the multi-scale compactification , and intersection-theoretic counting—to compute the degree of the residue map, including a closed formula and its resonant refinements via . The paper also provides flat-geometric proofs in special cases, derives an alternative gamma-function expression, and analyzes degenerations and resonance to describe ramification along resonance loci. As an application, it counts cone spherical metrics with dihedral monodromy, yielding explicit numbers in terms of the orders of the singularities. Overall, the work extends the abelian () results to higher , linking residue data, intersection theory, and flat-geometry in the study of -differentials.

Abstract

Consider the strata of primitive -differentials on the Riemann sphere whose singularities, except for two, are poles of order divisible by . The map that assigns to each -differential the -residues at these poles is a ramified cover of its image. Generalizing results known in the case of abelian differentials, we describe the ramification locus of this cover and provide a formula, involving the -factorial function, for the cardinality of each fiber. We prove this formula using intersection calculations on the multi-scale compactification of the strata of -differentials. In special cases, we also give alternative proofs using flat geometry. Finally, we present an application to cone spherical metrics with dihedral monodromy.

Paper Structure

This paper contains 28 sections, 20 theorems, 97 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper:
  3. Acknowledgements.
  4. $k$-differentials
  5. $k$-differentials and local models of singularities
  6. Strata of $k$-differentials
  7. $(1/k)$-translation structures
  8. Resonance stratification
  9. Periods of saddle connections
  10. Resonance stratification of the residual space
  11. The residual systole and flat continuation
  12. The multi-scale compactification of isoresidual loci
  13. The multi-scale compactification of strata
  14. The closure of loci of $k$-differentials with linear residue conditions
  15. The closure of isoresidual fibers
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $k\geq2$ and $\mu=(a_1,a_2,-b_1,\ldots,-b_p)$ be a partition of $-2k$ such that $b_{1},\dots,b_{p}$ are positive integer multiples of $k$, while $a_{1}$ and $a_{2}$ are coprime to $k$. The isoresidual map from the stratum $\Omega^{k}\mathcal{M}_{0}(\mu)$ of (marked) $k$-differentials on $\mathbb where for any $I\subset \{1,\ldots,p\}$

Figures (5)

  • Figure 1: The level graphs $\Gamma$ and $\Gamma'$, illustrating the $\mathcal{E}_{\Lambda}^{k}$-GRC. Vertices at level $\infty$ are shown in white.
  • Figure 2: The graph in $\mathcal{G}_{r,I}$ with $I=\lbrace 1 \rbrace \cup \lbrace 2,3 \rbrace$.
  • Figure 3: Quadratic differentials in the stratum ${\Omega^{2}\mathcal{M}}_{0}(3,-1;-2,-2,-2)$ with generic residues.
  • Figure 4: Quadratic differentials in the stratum ${\Omega^{2}\mathcal{M}}_{0}(1,1;-2,-2,-2)$ with generic residues.
  • Figure 5: Quadratic differentials in the stratum ${\Omega^{2}\mathcal{M}}_{0}(1,3;-2,-2,-2,-2)$ with generic residues.

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • ...and 38 more