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Area Siegel--Veech constants for affine invariant submanifolds of REL zero

Dawei Chen, Elise Goujard, Martin Möller

TL;DR

This work advances the understanding of area Siegel--Veech constants for affine invariant submanifolds by linking them to volumes of principal boundary strata in the multi-scale compactification, and by proving the CMS23a conjecture in the REL zero case. The authors develop the notion of principal graphs to capture the dominant boundary contributions and show that $c_{\text{area}}(\mathcal{M})$ can be computed as a boundary-intersection number, $c_{\text{area}}(\mathcal{M}) = -\frac{1}{4\pi^2} \cdot \frac{\int_{\mathbb{P}\overline{\mathcal{M}}} \xi^{m-1} \delta}{\int_{\mathbb{P}\overline{\mathcal{M}}} \xi^{m}}$, for REL zero. The paper then provides explicit boundary-graph analyses for several strata, including strata of quadratic differentials with odd zeros and the gothic locus, delivering concrete constants and their boundary decompositions (e.g., $c_{\text{area}}(\Omega\mathcal{G}) = \frac{49}{13\pi^2}$ and per-type contributions). The methods combine detailed volume computations, boundary-disintegration techniques, and intersection theory to connect flat-geometry growth rates with algebro-geometric data, offering a robust framework for evaluating area SV constants across REL-zero examples and suggesting pathways for general REL cases.

Abstract

We describe the principal boundary of an arbitrary affine invariant submanifold of REL zero in terms of level graphs of the multi-scale compactification of strata of Abelian differentials with prescribed orders of zeros. We show that the area Siegel--Veech constant of the affine invariant submanifold can be obtained by using volumes of the principal boundary strata. As an application, we prove the conjectural formula in [CMS23a] that computes the area Siegel--Veech constant via intersection theory in the case of REL zero. In particular, the formula holds for strata of quadratic differentials with odd orders of zeros and for the gothic locus. We also explicitly describe the principal boundary components of the gothic locus and their individual contributions to the area Siegel--Veech constant

Area Siegel--Veech constants for affine invariant submanifolds of REL zero

TL;DR

This work advances the understanding of area Siegel--Veech constants for affine invariant submanifolds by linking them to volumes of principal boundary strata in the multi-scale compactification, and by proving the CMS23a conjecture in the REL zero case. The authors develop the notion of principal graphs to capture the dominant boundary contributions and show that can be computed as a boundary-intersection number, , for REL zero. The paper then provides explicit boundary-graph analyses for several strata, including strata of quadratic differentials with odd zeros and the gothic locus, delivering concrete constants and their boundary decompositions (e.g., and per-type contributions). The methods combine detailed volume computations, boundary-disintegration techniques, and intersection theory to connect flat-geometry growth rates with algebro-geometric data, offering a robust framework for evaluating area SV constants across REL-zero examples and suggesting pathways for general REL cases.

Abstract

We describe the principal boundary of an arbitrary affine invariant submanifold of REL zero in terms of level graphs of the multi-scale compactification of strata of Abelian differentials with prescribed orders of zeros. We show that the area Siegel--Veech constant of the affine invariant submanifold can be obtained by using volumes of the principal boundary strata. As an application, we prove the conjectural formula in [CMS23a] that computes the area Siegel--Veech constant via intersection theory in the case of REL zero. In particular, the formula holds for strata of quadratic differentials with odd orders of zeros and for the gothic locus. We also explicitly describe the principal boundary components of the gothic locus and their individual contributions to the area Siegel--Veech constant

Paper Structure

This paper contains 23 sections, 17 theorems, 91 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Affine invariant submanifolds and their closures
  3. Affine invariant submanifolds
  4. The multi-scale compactification
  5. The closures of affine invariant submanifolds
  6. Siegel--Veech constants and principal graphs
  7. Siegel--Veech constants
  8. Volume normalization
  9. Principal graphs
  10. Principal boundary but higher codimension
  11. Coordinates adapted to the principal boundary
  12. The parameter space for cylinders
  13. Components of the principal boundary
  14. Disintegration of the volume forms
  15. Siegel--Veech constants as intersection numbers
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\mathcal{M}$ be an affine invariant submanifold of REL zero and complex dimension $m+1$. Then the area Siegel--Veech constant of $\mathcal{M}$ is equal to

Figures (6)

  • Figure 1: The level graphs $\Gamma_1$ (left) and $\Gamma_2$ (middle) and parameters on a flat surface in $\Omega {\mathcal{M}}_2(2)$ (right)
  • Figure 2: Integration domain in the variables $w$ and $H$
  • Figure 3: The principal graph of the minimal stratum $\mathcal{H}(2g-2)$
  • Figure 4: A flat surface parameterized in the locus of cyclic forms inside the gothic locus (from MMWgothic)
  • Figure 5: The top-level component of Type I and a regluing (sides with the same symbol $\alpha_j$ being identified)
  • ...and 1 more figures

Theorems & Definitions (34)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2: BDG
  • proof
  • Definition 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • ...and 24 more