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Connected components of generalized strata of meromorphic differentials with residue conditions

Myeongjae Lee, Yiu Man Wong

TL;DR

The paper classifies the connected components of generalized strata \\mathcal{P}(\\mu^{\\mathfrak{R}}) of meromorphic differentials with prescribed residue conditions by translating the problem to the study of one-dimensional bases and their principal boundaries. It introduces a robust framework based on the multi-scale compactification, enhanced level graphs, and plumbing to decompose strata into simpler boundary pieces, then performs an inductive analysis across dimensions. The main contributions are a complete invariant-based classification (hyperellipticity, ramification profile, spin parity, rotation number, and index) and a detailed map of how non-hyperelliptic and hyperelliptic components relate through surgeries like breaking up a zero, bubbling a handle, and simple-pole operations, with explicit boundary types (Type I, II, III) and equatorial-arc dynamics. The results illuminate the topology of the boundary complex of the strata’s compactification and pave the way for understanding irreducible components of boundary strata in multi-scale settings, with extensions to genus-zero/one residueless cases and hyperelliptic/non-hyperelliptic dichotomies.

Abstract

Generalized strata of meromorphic differentials are loci within the usual strata of differentials where certain sets of residues sum to zero. They naturally appear in the boundary of the multi-scale compactification of the usual strata. The classification of generalized strata is a key step towards understanding the irreducible components of the boundary strata of the multi-scale compactification. The connected components of generalized strata of residueless differentials were classified in \cite{lee2023connected}. In the present paper, we classify the connected components of generalized strata of meromorphic differentials in full generality.

Connected components of generalized strata of meromorphic differentials with residue conditions

TL;DR

The paper classifies the connected components of generalized strata \\mathcal{P}(\\mu^{\\mathfrak{R}}) of meromorphic differentials with prescribed residue conditions by translating the problem to the study of one-dimensional bases and their principal boundaries. It introduces a robust framework based on the multi-scale compactification, enhanced level graphs, and plumbing to decompose strata into simpler boundary pieces, then performs an inductive analysis across dimensions. The main contributions are a complete invariant-based classification (hyperellipticity, ramification profile, spin parity, rotation number, and index) and a detailed map of how non-hyperelliptic and hyperelliptic components relate through surgeries like breaking up a zero, bubbling a handle, and simple-pole operations, with explicit boundary types (Type I, II, III) and equatorial-arc dynamics. The results illuminate the topology of the boundary complex of the strata’s compactification and pave the way for understanding irreducible components of boundary strata in multi-scale settings, with extensions to genus-zero/one residueless cases and hyperelliptic/non-hyperelliptic dichotomies.

Abstract

Generalized strata of meromorphic differentials are loci within the usual strata of differentials where certain sets of residues sum to zero. They naturally appear in the boundary of the multi-scale compactification of the usual strata. The classification of generalized strata is a key step towards understanding the irreducible components of the boundary strata of the multi-scale compactification. The connected components of generalized strata of residueless differentials were classified in \cite{lee2023connected}. In the present paper, we classify the connected components of generalized strata of meromorphic differentials in full generality.
Paper Structure (57 sections, 85 theorems, 168 equations, 15 figures, 9 tables)

This paper contains 57 sections, 85 theorems, 168 equations, 15 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Main theorems
  3. Structure of the paper
  4. Motivation and future work
  5. Deformation of flat surfaces and principal boundary of strata
  6. Period coordinates of the strata of differentials
  7. Flat surfaces with degenerate core
  8. Zero-dimensional strata
  9. The multi-scale compactification of strata
  10. Plumbing construction
  11. Case (i):
  12. Case (ii):
  13. Configuration of Saddle Connections and Principal Boundary
  14. Level graphs of principal boundary
  15. Existence of principal boundaries of certain Type
  16. ...and 42 more sections

Key Result

Theorem 1.4

There exists a one-to-one correspondence between the set of hyperelliptic components of a stratum $\mathcal{P}(\mu^\mathfrak{R})$ of positive dimension and the set of ramification profiles of $\mathcal{P}(\mu^\mathfrak{R})$.

Figures (15)

  • Figure 1: Example (left) and non-example (right) of multi-curves $\mathfrak{R}$-homologous to zero
  • Figure 2: Type I and II polar domains and their representation by basic domains
  • Figure 3: breaking up a zero
  • Figure 4: Plumbing two nodes
  • Figure 5: Modification of top level flat surface case (i)
  • ...and 10 more figures

Theorems & Definitions (170)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 160 more