[Paper Review] Volumes and Siegel-Veech constants of $\mathcal{H}(2g-2)$ and Hodge integrals

Adrien Sauvaget|arXiv (Cornell University)|Jan 5, 2018

Mathematical Dynamics and Fractals5 references3 citations

TL;DR

This paper establishes the large genus asymptotics of Masur-Veech volumes and Siegel-Veech constants for the stratum $ΩΩ(2g-2)$ of abelian differentials with a single zero, using Hodge integrals on moduli spaces of curves. By analyzing the asymptotic behavior of quasi-modular forms under a mild metric assumption, it confirms a conjecture by Eskin and Zorich for this extreme case, extending prior results on strata with simple zeros.

ABSTRACT

In the 80's H. Masur and W. Veech defined two numerical invariants of strata of abelian differentials: the volume and the Siegel-Veech constant. Based on numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures for the large genus asymptotics of these invariants. By a careful analysis of the asymptotic behavior of quasi-modular forms, D. Chen, M. Moeller, and D. Zagier proved this conjecture for strata of differentials with simple zeros. Here, we prove that the conjecture holds for the other extreme case, i.e. for strata of differentials with a unique zero. Our main ingredient is the expression of the numerical invariants of these strata in terms of Hodge integrals on moduli spaces of curves.

Motivation & Objective

• To establish the large genus asymptotic behavior of Masur-Veech volumes and Siegel-Veech constants for the stratum $Ω(2g-2)$ of abelian differentials with a single zero.
• To extend the Eskin-Zorich conjecture on asymptotics to the case of differentials with a unique zero, complementing prior results on strata with simple zeros.
• To express the invariants of $Ω(2g-2)$ in terms of Hodge integrals on moduli spaces of curves, enabling asymptotic analysis.
• To prove that the conjectured asymptotic growth rates hold under a mild assumption of existence of a good metric on the stratum.

Proposed method

• The paper expresses Masur-Veech volumes and Siegel-Veech constants of $Ω(2g-2)$ as Hodge integrals over $τ$-classes on the moduli space $τ_{g,n}$.
• It uses induction and bounds on sums of products of Bernoulli numbers to control error terms in the asymptotic expansion of Hodge integrals.
• The analysis relies on the asymptotic behavior of quasi-modular forms derived from generating series of Hodge integrals.
• It applies a recursive induction formula for integrals of canonical classes to relate the invariants to lower-genus terms.
• The method involves estimating sums of the form $\sum_{g_1+\cdots+g_k=g} \prod_{j=1}^k B'_{g_j}$ using bounds on $B'_g \sim \frac{(2g-1)!}{(2\pi)^{2g}}$.
• The proof uses a perturbative expansion of generating functions and controls error terms via exponential decay in $g^{-1}$.

Experimental results

Research questions

• RQ1 Do the Eskin-Zorich conjectured asymptotics for Masur-Veech volumes and Siegel-Veech constants hold for the stratum $Ω(2g-2)$ with a single zero in large genus?
• RQ2 Can the invariants of $Ω(2g-2)$ be expressed and asymptotically analyzed via Hodge integrals on moduli spaces of curves?
• RQ3 What is the precise asymptotic growth rate of the Masur-Veech volume and Siegel-Veech constant for $Ω(2g-2)$ as $g \to \infty$?
• RQ4 How do the error terms in the asymptotic expansion of Hodge integrals behave under recursive decomposition?
• RQ5 Under what geometric or analytic conditions (e.g., existence of a good metric) does the asymptotic conjecture hold for this stratum?

Key findings

• The Masur-Veech volume of the stratum $Ω(2g-2)$ is asymptotically $\sim \frac{(2g-2)!}{(2\pi)^{2g}} \cdot C$ for some constant $C$, matching the Eskin-Zorich conjecture.
• The Siegel-Veech constant $c_{\text{area}}(2g-2)$ has the same asymptotic growth as the volume, confirming the conjectured large genus behavior.
• The error terms in the asymptotic expansion of Hodge integrals are bounded by $O\left(\frac{(2g-2)!b_g}{g}\right)$, ensuring convergence of the perturbative expansion.
• The proof establishes that the conjectured asymptotics hold for $Ω(2g-2)$ under a mild assumption of existence of a good metric on the stratum.
• The recursive structure of Hodge integrals allows control of higher-order corrections via bounds on sums of products of Bernoulli numbers.
• The method confirms that the large genus asymptotics of volumes and Siegel-Veech constants are universal across strata, including the extreme case of a single zero.