Abstract
We state conjectures on the asymptotic behavior of the Masur–Veech volumes of strata in the moduli spaces of meromorphic quadratic differentials and on the asymptotics of their area Siegel–Veech constants as the genus tends to infinity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aggarwal, A.: Large genus asymptotics for volumes of strata of abelian differentials. J. Am. Math. Soc. (2019a). arXiv:1804.05431
Aggarwal, A.: Large genus asymptotics for Siegel–Veech constants. GAFA 29, 1295–1324 (2019b)
Athreya, J., Eskin, A., Zorich, A.: Right-angled billiards and volumes of moduli spaces of quadratic differentials on \({\mathbb{C}}{\rm P}^{1}\). Ann. Scient. ENS, 4ème série 49, 1307–1381 (2016)
Chen, D., Moeller, M.: Quadratic differentials in low genus: exceptional and non-varying strata. Ann. Sci. Éc. Norm. Supér. (4) 47(2), 309–369 (2014)
Chen, D., Moeller, M., Zagier, D.: Quasimodularity and large genus limits of Siegel–Veech constants. J. Am. Math. Soc. 31(4), 1059–1163 (2018)
Chen, D., Moeller, M., Sauvaget, A., Zagier, D.: Masur–Veech volumes and intersection theory on moduli spaces of abelian differentials (2019a). arXiv:1901.01785
Chen, D., Möller, M., Sauvaget, A.: With an appendix by G. Borot, A. Giacchetto, D. Lewanski, Masur–Veech volumes and intersection theory: the principal strata of quadratic differentials (2019b). arXiv:1912.02267
Delecroix, V., Goujard, E., Zograf, P., Zorich, A.: Masur–Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves (2019). arXiv:1908.08611
Eskin, A., Okounkov, A.: Asymptotics of number of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math. 145(1), 59–104 (2001)
Eskin, A., Zorich, A.: Volumes of strata of Abelian differentials and Siegel–Veech constants in large genera. Arnold Math. J. 1(4), 481–488 (2015)
Eskin, A., Masur, H., Zorich, A.: Moduli spaces of Abelian differentials: the principal boundary, counting problems, and the Siegel–Veech constants. Publ. l’IHES 97(1), 61–179 (2003)
Eskin, A., Kontsevich, M., Zorich, A.: Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow. Publ. l’IHES 120(1), 207–333 (2014)
Goujard, E.: Siegel–Veech constants for strata of moduli spaces of quadratic differentials. GAFA 25(5), 1440–1492 (2015a)
Goujard, E.: Table of volumes of strata of quadratic differentials (up to dimension 11) (2015b). https://www.math.u.bordeaux.fr/~egoujard/tablevoljune2015.pdf
Goujard, E.: Volumes of strata of moduli spaces of quadratic differentials: getting explicit values. Ann. Inst. Fourier 66(6), 2203–2251 (2016)
Kazarian, M.: Recursion for Masur–Veech volumes of moduli spaces of quadratic differentials (2019). arXiv:1912.10422
Lanneau, E.: Connected components of the strata of the moduli spaces of quadratic differentials. Ann. Sci. Éc. Norm. Supér. (4) 41(1), 1–56 (2008)
Masur, H.: Interval exchange transformations and measured foliations. Annals of Math. 115, 169–200 (1982)
Masur, H., Zorich, A.: Multiple saddle connections on flat surfaces and principal boundary of the moduli spaces of quadratic differentials. GAFA 18, 919–987 (2008)
Sauvaget, A.: Volumes and Siegel–Veech constants of \({\cal{H}}(2g-2)\) and Hodge integrals. GAFA 28(6), 1756–1779 (2018)
Sauvaget, A.: The large genus asymptotic expansion of Masur–Veech volumes. IMRN (2019). arXiv:1903.04454
Veech, W.: Gauss measures for transformations on the space of interval exchange maps. Ann. Math. 115, 201–242 (1982)
Yang, D., Zagier, D., Zhang, Y.: Asymptotics of the Masur–Veech volumes (2020) (in preparation)
Zorich, A.: Square Tiled Surfaces and Teichmüller Volumes of the Moduli Spaces of Abelian Differentials. Rigidity in Dynamics and Geometry (Cambridge, 2000), pp. 459–471, Springer, Berlin (2002)
Zorich, A.: Asymptotic values of Siegel—Veech constants, Appendix to the paper of A. Aggarwal Large Genus Asymptotics for Volumes of Strata of Abelian Differentials. J. Am. Math. Soc. (2019). arXiv:1804.05431
Acknowledgements
We thank D. Chen, M. Moeller and A. Sauvaget for their formula and for their interest to the conjectural large genus asymptotics described above. We express special thanks to D. Chen for a list of typos found in the first version of this paper. We are very much indebted to M. Kazarian for his very efficient recursive formula for the linear Hodge integrals providing a reliable test of our conjectures. We thank the anonymous referee for helpful comments which allowed to improve the presentation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
A. Aggarwal is supported by the NSF Graduate Research Fellowship under grant number DGE1144152 and NSF grant DMS-1664619. E. Goujard was partially supported by PEPS. The research of Section 2 was conducted at Saint Petersburg State University under support of the RSF grant 19-71-30002.
Rights and permissions
About this article
Cite this article
Aggarwal, A., Delecroix, V., Goujard, É. et al. Conjectural Large Genus Asymptotics of Masur–Veech Volumes and of Area Siegel–Veech Constants of Strata of Quadratic Differentials. Arnold Math J. 6, 149–161 (2020). https://doi.org/10.1007/s40598-020-00139-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40598-020-00139-7