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

TL;DR

This paper proves a conjecture relating volumes and Siegel-Veech constants of certain strata of abelian differentials with a unique zero, using Hodge integrals and asymptotic analysis.

Contribution

It establishes the conjecture for strata with a single zero, extending previous results to this extreme case using Hodge integrals.

Findings

Conjecture holds for strata with a unique zero.

Involves asymptotic analysis of quasi-modular forms.

Expresses invariants in terms of Hodge integrals.

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.