Volumes and Siegel–Veech constants of $${\mathcal{H}}$$H (2G − 2) and 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 that this conjecture holds for strata of differentials with simple zeros.Here, with a mild assumption of existence of a good metric, we show 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.

Bill Veech's contributions to dynamical systems

“…An analysis of the generating functions leads to a proof (for principal strata) of Eskin-Zorich conjectures on large genus asymptotics for the volume and the Siegel-Veech constants. A. Sauvaget [68] has proved similar results in the case of strata of Abelian differentials with a single zero.…”

Large Genus Asymptotics for Siegel–Veech Constants

“…The former prediction was based on numerical data provided by a program written by Eskin that implements the algorithm of Eskin-Okounkov to evaluate volumes of strata of genus g ≤ 10. This prediction was established in the cases of the principal and minimal strata (see below) in the works of Chen-Möller-Zagier [3] and Sauvaget [17], respectively; it was later confirmed in general in [1]. These results were recently used by Masur-Rafi-Randecker [14] to analyze the diameter of a generic translation surface in the minimal stratum of large genus.…”

“…For example, in the similar context of Weil-Petersson volumes, such questions were investigated at length in [15,16,23], and they were also considered in algebraic geometry to understand slopes of Teichmüller curves in [2]. From the perspective of flat geometry, large genus asymptotics have also been studied in a number of recent works [1,3,10,14,17].…”

Large genus asymptotics for volumes of strata of abelian differentials

“…This completes the proof of the first part of the statement. By the formula of Vorobets (see (2.16) in [7] or the original paper [21]), the Siegel-Veech constant c area (H(m)) is expressed in terms of the Siegel-Veech constants of configurations of homologous closed saddle connections as follows (18) c area (H(m)) = 1 dim C H(m) − 1 · g−1 q=1 q · Conf igurations C containing q cylinders c C (H(m)) .…”

“…Second, the work of Sauvaget [18] established (1.1) in the case of the minimal stratum m = (2g − 2), when ω has one zero with multiplicity 2g − 2. Through an analysis of Hodge integrals on the moduli space of curves (based on his earlier work [17]), he shows as Theorem 1.9 of [18] that…”