Large genus asymptotics for volumes of strata of abelian differentials

Abstract

In this paper we consider the large genus asymptotics for Masur-Veech volumes of arbitrary strata of Abelian differentials. Through a combinatorial analysis of an algorithm proposed in 2002 by Eskin-Okounkov to exactly evaluate these quantities, we show that the volumeThis confirms a prediction of Eskin-Zorich and generalizes some of the recent results of Chen-Möller-Zagier and Sauvaget, who established these limiting statements in the special cases m = 1 2g−2 and m = (2g − 2), respectively.We also include an Appendix by Anton Zorich that uses our main result to deduce the large genus asymptotics for Siegel-Veech constants that count certain types of saddle connections.

Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials

“…When the stratum is disconnected, we also show that the theorem holds for each connected component under Assumption 6.1. We remark that as an asymptotic equality as g tends to infinity, the formula (7) for the entire stratum was previously shown in the appendix by Zorich to [3] for saddle connections of multiplicity one and by Aggarwal [4] for all multiplicities.…”

“…where ξ is the universal line bundle class of the projectivized Hodge bundle and ψ j is the vertical cotangent line bundle class associated to the jth marked point (see Sect. 3 for a more precise definition of these tautological classes).…”

Uniform Lower Bound for Intersection Numbers of ψ-Classes

“…. , d n ); the second equality is the string equation (2); the inequality in the middle of the second line is the induction assumption; the equality in the beginning of the third line is definition (17) of (1+δ dilaton (1, d)); the inequality in the beginning of the last line is a direct implication of (28); the last inequality is an implication of (23). Suppose now that k ≥ 1.…”

“…, d n−s . The second equality is the string equation (2). (Recall that by convention, if one of d n−s−1 or d n−s is equal to zero, the term, containing the negative index d n−s−1 − 1 or d n−s − 1 respectively, is missing in the string equation and below.)…”

“…Certain universality phenomena in flat and hyperbolic geometry and in dynamics of surfaces manifest themselves in large genera. The large genus asymptotics of the Masur-Veech volumes of strata in moduli spaces of Abelian differentials conjectured in [12] was successfully proved by independent methods in [2] and in [6]. However, the analogous conjectures stated in [4] on the large genus asymptotics of the Masur-Veech volumes of strata in moduli spaces of quadratic differentials are open.…”