zbMATH — the first resource for mathematics

Towards large genus asymptotics of intersection numbers on moduli spaces of curves. (English) Zbl 1327.14136
Let \(V_{g,n}\) be the Weil-Petterson volume of the moduli space \(\mathcal{M}_{g,n}\) of complex algebraic curves of genus \(g\) with \(n\) punctures. The paper under review studies the asymptotic behaviour of \(V_{g,n}\) as \(g \rightarrow \infty\). The main result is theorem 1.2, which states that there exists a constant \(C \in (0,\infty)\), such that for any given \(k \geq 1, n \geq 0\),
\[ V_{g,n} = C \frac {(2g-3+n)! (4 \pi^2)^{2g-3+n}} {\sqrt g} \left(1 + \sum_{i=1}^{k} \frac{c_n^{(i)}} {g^i} + O\left(\frac {1} {g^{k+1}}\right)\right) \]
as \(g \rightarrow \infty\). Each \(c_n^{(i)}\) is a polynomial in \(n\) of degree \(2i\) with coefficients in \(\mathbb{Q}[\pi ^{-2}, \pi ^2]\), effectively computable.
This result is consistent with the conjecture made by the second author in [“On the large genus asymptotics of Weil-Peterson volumes”, Preprint (2008) arXiv:0812.0544]. In fact, the conjecture being true, the constant \(C\) would be \(\frac {1}{\sqrt \pi}\).
The proof of the result is made by studying the quotient \(\frac {V_{g+1,n}} {V_{g,n}}\) for a given \(n\), and \(g \rightarrow \infty\). Estimates for this ratio were obtained by the first author in [J. Differ. Geom. 94, No. 2, 267–300 (2013; Zbl 1270.30014)].
In the final section of the paper, the authors consider the behaviour of \(V_{g,n(g)}\) when \(n(g) \rightarrow \infty\) as \(g \rightarrow \infty\) but \(n(g)^2/g \rightarrow 0\).

14H10 Families, moduli of curves (algebraic)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
Full Text: DOI arXiv
[1] Arbarello, E.; Cornalba, M., Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, Journal of Algebraic Geometry, 5, 705-709, (1996) · Zbl 0886.14007
[2] Do, N.; Norbury, P., Weil-Petersson volumes and cone surfaces, Geometriae Dedicata, 141, 93-107, (2009) · Zbl 1177.32008
[3] H.M. Edwards. Riemann’s Zeta Function. Academic Press, New York (1974). · Zbl 0315.10035
[4] Eynard, B., Recursion between Mumford volumes of moduli spaces, Annales Henri Poincaré, 12, 1431-1447, (2009) · Zbl 1245.14013
[5] B. Eynard and N. Orantin. Invariants of algebraic curves and topological expansion. Communications in Number Theory and Physics, 1:2 (2007), 347-452. · Zbl 1161.14026
[6] Grushevsky, S., An explicit upper bound for Weil-Petersson volumes of the moduli spaces of punctured Riemann surfaces, Mathematische Annalen, 321, 1-13, (2001) · Zbl 0998.32008
[7] J. Harris and I. Morrison. Moduli of curves. In: Graduate Texts in Mathematics, Vol. 187. Springer, New York (1998). · Zbl 0913.14005
[8] Kaufmann, R.; Manin, Y.; Zagier, D., Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves, Communications in Mathematical Physics, 181, 736-787, (1996) · Zbl 0890.14011
[9] M.E. Kazarian. (2006). (Private communication).
[10] Kazarian, M.E., Lando S.K. an algebro-geometric proof of witten’s conjecture, Journal of the American Mathematical Society, 20, 1079-1089, (2007) · Zbl 1155.14004
[11] Kontsevich, M., Intersection on the moduli space of curves and the matrix Airy function, Communications in Mathematical Physics, 147, 1-23, (1992) · Zbl 0756.35081
[12] Liu, K.; Xu, H., Recursion formulae of higher Weil-Petersson volumes, International Mathematics Research Notices IMRN, 5, 835-859, (2009) · Zbl 1186.14059
[13] Liu, K.; Xu, H., Mirzakharni’s recursion formula is equivalent to the Witten-Kontsevich theorem, Asterisque, 328, 223-235, (2009) · Zbl 1194.14040
[14] Yu. Manin and P. Zograf. Invertible cohomological field theories and Weil-Petersson volumes. Annales de l’institut Fourier, 50:2 (2000), 519-535. · Zbl 1001.14008
[15] McShane, G., Simple geodesics and a series constant over Teichmüller space, Inventiones mathematicae, 132, 607-632, (1998) · Zbl 0916.30039
[16] M. Mirzakhani. Weil-Petersson volumes and intersection theory on the moduli space of curves. Journal of the American Mathematical Society, 20:1 (2007), 1-23. · Zbl 1120.32008
[17] Mirzakhani, M., Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Inventiones mathematicae, 167, 179-222, (2007) · Zbl 1125.30039
[18] Mirzakhani, M., Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus, Journal of Differential Geometry, 94, 267-300, (2013) · Zbl 1270.30014
[19] Mulase, Y.; Safnuk, P., Mirzakhani’s recursion relations, Virasoro constraints and the KdV hierarchy, Indian Journal of Mathematics, 50, 189-228, (2008) · Zbl 1144.14030
[20] A. Okounkov and R. Pandharipande. Gromov-Witten theory, Hurwitz numbers, and matrix models. Proceedings of Symposia in Pure Mathematics, 80.1 (2009), 325-414. · Zbl 1205.14072
[21] Penner, R., Weil-Petersson volumes, Journal of Differential Geometry, 35, 559-608, (1992) · Zbl 0768.32016
[22] Schumacher, G.; Trapani, S., Estimates of Weil-Petersson volumes via effective divisors, Communications in Mathematical Physics, 222, 1-7, (2001) · Zbl 0988.32013
[23] Witten, E., Two-dimensional gravity and intersection theory on moduli spaces, Surveys in Differential Geometry, 1, 243-269, (1991) · Zbl 0757.53049
[24] S. Wolpert. On the homology of the moduli of stable curves. Annals of Mathematics, 118:2 (1983), 491-523 · Zbl 0575.14024
[25] P. Zograf. On the large genus asymptotics of Weil-Petersson volumes (2008). (arXiv:0812.0544).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.