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Volumes and Siegel–Veech constants of \({\mathcal{H}}\) (2G − 2) and Hodge integrals

  • Adrien Sauvaget
Article
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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.

Keywords and phrases

Moduli space of curves Translation surfaces Masur–Veech volumes Hodge integrals 

Mathematics Subject Classification

14H15 14N10 30F30 30F60 14C17 

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Notes

Acknowledgements

I would like to thank Dimitri Zvonkine, Martin Moëller, Dawei Chen, Xavier Blot, Siarhei Finski and Felix Janda for very useful conversations on intersection of tautological classes over spaces of differentials. I am also very thankful to Anton Zorich and Charles Fougeron for having introduced me to the topic of large-genus invariants (and for having provided tables of numerical computations that allowed me to understand part of the results of the paper). Finally, I am very grateful to Elie de Panafieu for his precious help to handle the asymptotic analysis.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.IMJ-PRGUniversité Pierre et Marie CurieParisFrance

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