Asymptotic distributions of the Betti and Hodge numbers of ℳ̅g,n
Abstract
We study the asymptotic distributions of the Betti and Hodge numbers of the moduli spaces ℳ̅g,n of stable pointed curves and the all-light Hassett spaces ℳ̅g,0|n, for fixed genus g≥1 as the number of markings tends to infinity. We prove central limit theorems for the full Betti distributions, whose variances grow linearly in n with leading coefficients independent of the genus. For the normalized even Betti numbers, we establish refined local limit theorems with estimates uniform in degree. These local estimates yield asymptotic formulas for individual even Betti numbers and determine the sharp central ultra-log-concavity thresholds (3(e−2))/(2(3−e)) and $3$, respectively. For g≥2, the off-diagonal Hodge components form proportions of exact orders n−1/2 and n−1 of the total cohomology of the two spaces, respectively. The conditional distributions of the Hodge difference p−q given p≠ q converge, for both spaces, to a single discrete law independent of the genus. We also prove that tautological classes account for all but a proportion of exact order n−1 of the even cohomology of ℳ̅g,n. The tautological degree distribution converges to the standard Gaussian under the centering and scaling of the ambient even Betti distribution. An explicit asymptotic formula for the ratio of the ordinary Euler characteristics of the light and heavy spaces answers a question of Kannan, Serpente and Yun. The proofs combine the heavy/light substitution with a stratification of the light spaces by stable cores, singularity analysis of Hodge generating functions, and refined Fourier estimates exploiting Poincaré duality.
Related
- Invariants of big classes from asymptotic total Betti numbers
- Betti Numbers of Odd Biisotropic Symplectic Grassmannians
- Graded Betti numbers of graded Möbius algebras of uniform matroids
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- ℓ2-Betti numbers and Σ-invariants of graph products of cyclic groups and their automorphisms
BibTeX
@misc{choi2026asymptotic,
title = {{Asymptotic distributions of the Betti and Hodge numbers of $\overline\mathcal M_g,n$}},
author = {Jinwon Choi and Young-Hoon Kiem},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38161},
eprint = {2610.38161},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38161},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Asymptotic distributions of the Betti and Hodge numbers of ℳ̅_g,n
AU - Jinwon Choi
AU - Young-Hoon Kiem
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38161
UR - https://arxiv.org/abs/2610.38161
AB - We study the asymptotic distributions of the Betti and Hodge numbers of the moduli spaces ℳ̅_g,n of stable pointed curves and the all-light Hassett spaces ℳ̅_g,0|n, for fixed genus g≥1 as the number of markings tends to infinity. We prove central limit theorems for the full Betti distributions, whose variances grow linearly in n with leading coefficients independent of the genus. For the normalized even Betti numbers, we establish refined local limit theorems with estimates uniform in degree. These local estimates yield asymptotic formulas for individual even Betti numbers and determine the sharp central ultra-log-concavity thresholds (3(e−2))/(2(3−e)) and $3$, respectively. For g≥2, the off-diagonal Hodge components form proportions of exact orders n^−1/2 and n^−1 of the total cohomology of the two spaces, respectively. The conditional distributions of the Hodge difference p−q given p≠ q converge, for both spaces, to a single discrete law independent of the genus. We also prove that tautological classes account for all but a proportion of exact order n^−1 of the even cohomology of ℳ̅_g,n. The tautological degree distribution converges to the standard Gaussian under the centering and scaling of the ambient even Betti distribution. An explicit asymptotic formula for the ratio of the ordinary Euler characteristics of the light and heavy spaces answers a question of Kannan, Serpente and Yun. The proofs combine the heavy/light substitution with a stratification of the light spaces by stable cores, singularity analysis of Hodge generating functions, and refined Fourier estimates exploiting Poincaré duality.
ER -