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Relative period realization of holomorphic differentials with prescribed invariants

Dawei Chen, Gianluca Faraco

TL;DR

This work resolves when a relative period representation χ can be realized in a fixed stratum by tying it to the known Haupt obstructions for absolute periods: if the absolute part χ_a is realizable in a stratum, then every compatible relative period χ is realizable in every connected component of that stratum. The authors distinguish two regimes: non-discrete χ_a, where zero-breaking and Schiffer moves yield isoperiodic deformations preserving holonomy and enabling componentwise realization, and discrete χ_a, where square-tiled surfaces arising from branched covers over the square torus are constructed to realize prescribed zeros and relative periods. Central innovations include systematic surgeries that preserve holonomy, a careful analysis of hyperelliptic and spin invariants during realizations, and explicit square-tiled/ origami-based constructions that realize representations in all components across connected strata. The results offer a complete answer to Simion Filip’s question and extend Haupt-type obstructions to relative-period realizations, with potential extensions to meromorphic differentials and broader invariants.

Abstract

We provide a complete description of realizable relative period representations for holomorphic differentials on Riemann surfaces with prescribed orders of zeros and additional invariants given by the hyperelliptic structure and spin parity. This answers a question posed by Simion Filip.

Relative period realization of holomorphic differentials with prescribed invariants

TL;DR

This work resolves when a relative period representation χ can be realized in a fixed stratum by tying it to the known Haupt obstructions for absolute periods: if the absolute part χ_a is realizable in a stratum, then every compatible relative period χ is realizable in every connected component of that stratum. The authors distinguish two regimes: non-discrete χ_a, where zero-breaking and Schiffer moves yield isoperiodic deformations preserving holonomy and enabling componentwise realization, and discrete χ_a, where square-tiled surfaces arising from branched covers over the square torus are constructed to realize prescribed zeros and relative periods. Central innovations include systematic surgeries that preserve holonomy, a careful analysis of hyperelliptic and spin invariants during realizations, and explicit square-tiled/ origami-based constructions that realize representations in all components across connected strata. The results offer a complete answer to Simion Filip’s question and extend Haupt-type obstructions to relative-period realizations, with potential extensions to meromorphic differentials and broader invariants.

Abstract

We provide a complete description of realizable relative period representations for holomorphic differentials on Riemann surfaces with prescribed orders of zeros and additional invariants given by the hyperelliptic structure and spin parity. This answers a question posed by Simion Filip.

Paper Structure

This paper contains 34 sections, 8 theorems, 33 equations, 20 figures, 1 table.

Table of Contents

  1. Introduction
  2. From the complex-analytic perspective to geometry
  3. The period map
  4. The strategy of proving the main Theorem
  5. Meromorphic differentials
  6. Structure of the paper
  7. Preliminaries of translation surfaces and period representations
  8. Strata of differentials
  9. Absolute and relative periods
  10. Translation surfaces
  11. Volume
  12. Further invariants
  13. Hyperelliptic involution
  14. Spin parity
  15. Surgeries on translation surfaces
  16. ...and 19 more sections

Key Result

Theorem A

Let $\chi\colon\textnormal{H}_1(S_{g},\,Z,\,\mathbb Z)\longrightarrow \mathbb{C}$ be a relative period representation with absolute period representation $\chi_a$. Let $\mu$ be any signature of holomorphic differential. If $\chi_a$ can be realized in $\mathcal{H}_g(\,\mu\,)$ then $\chi$ can be reali

Figures (20)

  • Figure 1: Labels of a slit
  • Figure 2: An $\varepsilon$-neighborhood of a zero of order $2$.
  • Figure 3: New labeling for breaking up a zero of order $2$ in two zeros of order $1$.
  • Figure 4: Movement of a zero of order $3$.
  • Figure 5: Bubbling a handle with positive volume.
  • ...and 15 more figures

Theorems & Definitions (43)

  • Theorem A
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: Translation surfaces
  • Lemma 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 33 more