Volumes and Siegel–Veech constants of H (2G − 2) and Hodge integrals

• Published in: Geometric and functional analysis Vol. 28; no. 6; pp. 1756 - 1779
• Main Author: Sauvaget, Adrien
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2018; Springer Nature B.V; Springer Verlag
• Subjects: Algebraic Geometry; Analysis; Asymptotic properties; Complex Variables; Integrals; Invariants; Mathematics; Mathematics and Statistics; Strata; Translation surfaces; 14N10; Hodge integrals; 14H15; Moduli space of curves; Masur–Veech volumes; 30F30; 30F60; 14C17; Masur-Veech volumes; Mathematics Subject Classification. 14H15; Mathematics Subject Classification. 14H15 14N10 30F30 30F60 14C17 Moduli space of curves translation surfaces Masur-Veech volumes Hodge integrals; 14C17 Moduli space of curves
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-018-0468-5

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.