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Volumes of moduli spaces of flat surfaces

Adrien Sauvaget

Abstract

We study the moduli spaces of flat surfaces with prescribed conical singularities. Veech showed that these spaces are diffeomorphic to the moduli spaces of marked Riemann surfaces, and endowed with a natural volume form depending on the orders of the singularities. We show that the volumes of these spaces are finite. Moreover we show that they are explicitely computable by induction on the Euler characteristics of the punctured surface for almost all orders of the singularities.

Volumes of moduli spaces of flat surfaces

Abstract

We study the moduli spaces of flat surfaces with prescribed conical singularities. Veech showed that these spaces are diffeomorphic to the moduli spaces of marked Riemann surfaces, and endowed with a natural volume form depending on the orders of the singularities. We show that the volumes of these spaces are finite. Moreover we show that they are explicitely computable by induction on the Euler characteristics of the punctured surface for almost all orders of the singularities.

Paper Structure

This paper contains 32 sections, 25 theorems, 134 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Veech's measure
  3. Moduli spaces of pluri-canonical divisors
  4. Flat recursion versus topological recursion
  5. From intersection theory to volumes
  6. Teichmüller space and the holonomy map
  7. Geodesic triangulations and parameters of $\widehat{{\mathcal{T}}}_{g,n}$
  8. $\mathbb{P} U(p,q,r)$-structure and volume forms
  9. Three compactifications of ${\mathcal{M}}(a,k)$
  10. Flat recursion formula
  11. Computation of (Masur-)Veech volumes
  12. Convergence for large $k$-values
  13. Wall-crossing properties of the flat recursion
  14. Flat surfaces with integral singularities
  15. Intersection theory of moduli spaces of $k$-differentials
  16. ...and 17 more sections

Key Result

Theorem 2

If $n\geq 2$, then there exists a rational piece-wise polynomial $\mathcal{V}\colon\Delta_{g,n}^+ \to \mathbb{R}$ of degree $4g-3+n$ such that the equality holds for all rational vectors $a$ without integral coordinates.

Figures (3)

  • Figure 1: Graphs of $a_1\mapsto (-1)^g{\mathcal{V}}(a_1,2g-a_1)$ (top), and $a_1\mapsto {\rm Vol}(a_1,2g-a_1)$ (bottom), in genus $g=1$ (left), and $g=2$ (right).
  • Figure 2: Polygonal representation of a surface in ${\mathcal{M}}((2/3,4/3),6)$ related by a cut-and-paste operation. If $r=e^{2i\pi/6}$, then the coordinates of this surface in the left representation are $z=(-r,1,r)$ and $e=(r,-r,1)$. We also represented the dual path to $\gamma_1$ in red.
  • Figure 3: Graphical representation of a star graph in ${\rm Star}_{7,3}^1$. The values $g_i$ are denoted at each vertex and the domain $\Delta_{\Gamma}(a)$ is the set of positive triples $(b_{1,1},b_{1,2},b_{2,1})$ satisfying $b_{2,1}=4-a_3,$ and $b_{1,1}+b_{1,2}=7-a_2$. It is empty if $a_2\geq 7$ or $a_3\geq 4$.

Theorems & Definitions (45)

  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1.1
  • proof
  • Proposition 1.2: Theorem 1.4 of CosMoeZac
  • Proposition 1.3
  • Theorem 1.4
  • Lemma 1.5
  • ...and 35 more