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Monodromy of Schwarzian equations with regular singularities

Gianluca Faraco, Subhojoy Gupta

TL;DR

This work provides a complete characterization of which PSL_2(C) representations arise as monodromies of Schwarzian equations with regular singularities on punctured finite-type surfaces S_{g,k}. The authors treat non-degenerate representations via a streamlined use of Fock–Goncharov coordinates and pleated-planes in ℍ^3, and they tackle degenerate (affine and dihedral) holonomies through explicit constructions of affine/translation structures and gluing along rays, including novel bubbling and grafting techniques. Consequences include a precise criterion for spherical cone-metrics and a broad result that every representation is realizable as a (branched) CP^1-structure with at most two branch points, yielding infinite fibers in many cases. The results significantly advance the understanding of the monodromy problem for punctured surfaces, connecting holonomy to both classical projective geometry and modern higher Teichmüller theory, with constructive methods that may be algorithmically implemented.

Abstract

Let $S$ be a punctured surface of finite type and negative Euler characteristic. We determine all possible representations $ρ:π_1(S) \to \text{PSL}_2(\mathbb{C})$ that arise as the monodromy of the Schwarzian equation on $S$ with regular singularities at the punctures. Equivalently, we determine the holonomy representations of complex projective structures on $S$, whose Schwarzian derivatives (with respect to some uniformizing structure) have poles of order at most two at the punctures. Following earlier work that dealt with the case when there are no apparent singularities, our proof reduces to the case of realizing a degenerate representation with apparent singularities. This mainly involves explicit constructions of complex affine structures on punctured surfaces, with prescribed holonomy. As a corollary, we determine the representations that arise as the holonomy of spherical metrics on $S$ with cone-points at the punctures.

Monodromy of Schwarzian equations with regular singularities

TL;DR

This work provides a complete characterization of which PSL_2(C) representations arise as monodromies of Schwarzian equations with regular singularities on punctured finite-type surfaces S_{g,k}. The authors treat non-degenerate representations via a streamlined use of Fock–Goncharov coordinates and pleated-planes in ℍ^3, and they tackle degenerate (affine and dihedral) holonomies through explicit constructions of affine/translation structures and gluing along rays, including novel bubbling and grafting techniques. Consequences include a precise criterion for spherical cone-metrics and a broad result that every representation is realizable as a (branched) CP^1-structure with at most two branch points, yielding infinite fibers in many cases. The results significantly advance the understanding of the monodromy problem for punctured surfaces, connecting holonomy to both classical projective geometry and modern higher Teichmüller theory, with constructive methods that may be algorithmically implemented.

Abstract

Let be a punctured surface of finite type and negative Euler characteristic. We determine all possible representations that arise as the monodromy of the Schwarzian equation on with regular singularities at the punctures. Equivalently, we determine the holonomy representations of complex projective structures on , whose Schwarzian derivatives (with respect to some uniformizing structure) have poles of order at most two at the punctures. Following earlier work that dealt with the case when there are no apparent singularities, our proof reduces to the case of realizing a degenerate representation with apparent singularities. This mainly involves explicit constructions of complex affine structures on punctured surfaces, with prescribed holonomy. As a corollary, we determine the representations that arise as the holonomy of spherical metrics on with cone-points at the punctures.

Paper Structure

This paper contains 31 sections, 40 theorems, 59 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Meromorphic projective structures
  4. Framed representations
  5. Proof of Theorem \ref{['thm1']}: Non-degenerate representations
  6. Fock-Goncharov coordinates
  7. Pleated planes in $\mathbb{H}^3$
  8. Constructing a projective structure
  9. Schwarzian derivative at the punctures
  10. Proof of Theorem \ref{['thm2']}: Affine holonomy with at least two punctures
  11. Trivial representation
  12. Translation structures
  13. Affine surfaces with at least two punctures
  14. Affine holonomy and a single puncture
  15. Necessary conditions
  16. ...and 16 more sections

Key Result

Theorem A

Let $\Pi$ be the fundamental group of a surface $S_{g,k}$ of genus $g$ and $k\geq 1$ punctures, where $2-2g-k<0$. A representation $\rho:\Pi \to {\mathrm{PSL}_2 (\mathbb{C})}$ arises as the monodromy of a $\mathbb{C}\mathrm{P}^1$-structure in $\mathcal{P}_g(k)$ if and only if one of the following ho

Figures (18)

  • Figure 1: The map $\Psi_\infty$ maps the grafted region $B(\tilde{e})$ to the "lune" $L_\alpha$ on $\mathbb{C}\mathrm{P}^1$ bounded by the circular arcs $\alpha_l$ and $\alpha_r$. The shaded regions on the right are the images of ${\Delta_l}$ and ${\Delta_r}$ under $\Psi^0_\infty$ (see §3.3).
  • Figure 2: The map $\Pi$ in Lemma \ref{['triv']} is a branched cover over the sphere with exactly three critical points and three branch-points.
  • Figure 3: The surface $S_{g,k}$ can be divided into a subsurface homeomorphic to $S_{g,1}$ (shown shaded) and its complement, homeomorphic to $S_{0,k+1}$.
  • Figure 4: A Dehn-twist changes the pair of handle-generators $\{\gamma_0, \gamma_1\}$ to the pair $\{\gamma_0, \gamma_1^\prime\}$ where $\gamma^\prime_1 = \gamma_0 + \gamma_1$ in homology.
  • Figure 5: Gluing $\Sigma_1$ and $\Sigma_2$ along rays results in a new surface $\Sigma$, see Definition \ref{['glue-ray']}.
  • ...and 13 more figures

Theorems & Definitions (90)

  • Definition 1.1
  • Definition 1.2
  • Theorem A
  • Corollary B
  • Theorem C
  • Corollary D
  • Corollary E
  • Definition 2.1
  • Definition 2.2: Branch-point
  • Definition 2.3
  • ...and 80 more