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Ends of the strata of differentials

Benjamin Dozier, Samuel Grushevsky, Myeongjae Lee

Abstract

We enumerate the ends of each stratum of meromorphic 1-forms on Riemann surfaces with prescribed multiplicities of zeroes and poles. Our proof uses degeneration techniques based on the construction by Bainbridge-Chen-Gendron-Grushevsky-Moeller of the moduli space of multi-scale differentials, together with recent classification of connected components of generalized strata by Lee-Wong. In particular, from these results we quickly deduce the theorem for holomorphic 1-forms, originally proved by Boissy.

Ends of the strata of differentials

Abstract

We enumerate the ends of each stratum of meromorphic 1-forms on Riemann surfaces with prescribed multiplicities of zeroes and poles. Our proof uses degeneration techniques based on the construction by Bainbridge-Chen-Gendron-Grushevsky-Moeller of the moduli space of multi-scale differentials, together with recent classification of connected components of generalized strata by Lee-Wong. In particular, from these results we quickly deduce the theorem for holomorphic 1-forms, originally proved by Boissy.
Paper Structure (4 sections, 10 theorems, 5 figures)

This paper contains 4 sections, 10 theorems, 5 figures.

Table of Contents

  1. Introduction
  2. Ends of strata of holomorphic 1-forms
  3. Ends of general meromorphic strata
  4. The horizontal irreducible boundary divisor in genus one

Key Result

Lemma 3

For any $\mathcal{H}^\circ$, if $\partial\overline{\mathcal{H}^\circ}$ is non-empty and connected, then $\mathcal{H}^\circ$ has exactly one end.

Figures (5)

  • Figure 1: Level graphs corresponding to a vertical degeneration that merges all the zeroes together. The arrow $\leftsquigarrow$ denotes the undegeneration of differentials; it points in the direction of the corresponding morphism of enhanced level graphs, while the differentials specialize in the opposite direction.
  • Figure 2: Level graphs corresponding to a degeneration of a point of $D_j^h$ to a point of $D_j^h\cap{D^{h,{\rm irr}}_{}}$.
  • Figure 3: Degeneration and undegeneration used in Case 2 to reduce to the case of a single top level vertex.
  • Figure 4: Degeneration and undegeneration used in Case 2 to reduce to the case of a single bottom level vertex.
  • Figure 6: Construction of $D_i$ for the case $m_i>1$.

Theorems & Definitions (26)

  • Remark 1
  • Remark 2
  • Lemma 3
  • proof
  • Theorem 4: BoissyEnds, the case of abelian differentials
  • proof : Proof of \ref{['thm:abelian']}
  • Claim A
  • Claim B
  • Claim C
  • Lemma 5
  • ...and 16 more