Conjectural Large Genus Asymptotics of Masur–Veech Volumes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an integer $g\ge2$, let $\widehat\Pi_{4g-4}$ consist of finite unordered lists $\boldsymbol d=(d_1,\ldots,d_n)$ with \[ d_i\in\{-1,0,1,2,\ldots\},\qquad\sum_i d_i=4g-4,\qquad \#\{i:d_i=-1\}\le\log g, \] where $\log$ is natural. The stratum $\mathcal Q_g(\boldsymbol d)$ parametrizes pairs $(X,q)$ consisting of a compact genus-$g$ Riemann surface and a meromorphic quadratic differential with the specified orders at its labeled singularities: $d_i=-1$ denotes a simple pole, $d_i>0$ a zero, and $d_i=0$ a marked regular point. Isomorphisms preserve the differential and labels. The quadratic stratum uses differentials that are not globally squares of Abelian differentials.

Fix the Masur--Veech normalization used by the sources. On the canonical orientation double cover $\widehat X$, where the pullback of $q$ is $\omega^2$, use anti-invariant relative periods of $\omega$. The period-coordinate lattice consists of linear forms taking values in $\mathbb Z+i\mathbb Z$ on anti-invariant integral relative homology, and its covolume is $1$. The relative marked set consists of preimages of zeros and marked regular points. Put $\mathcal Q_{g,1}(\boldsymbol d)=\{(X,q):\int_X|q|=1/2\}$. If $D=2g+n-2$ is the complex dimension, define $\mu_1$ by \[ d\mu=r^{2D-1}\,dr\,d\mu_1,\qquad q=r^2q_1,\quad q_1\in\mathcal Q_{g,1}(\boldsymbol d), \] and $\operatorname{Vol}\mathcal Q_g(\boldsymbol d)=\int_{\mathcal Q_{g,1}(\boldsymbol d)}d\mu_1$. In particular zeros are labeled; adding a labeled regular point multiplies the volume by $2$. Only nonempty strata enter the questions.

Statement recorded in the supplied source.

Establish the uniform large-genus asymptotic formula \[ \operatorname{Vol}\mathcal Q_g(\boldsymbol d) =\frac4\pi\prod_{i=1}^{n}\frac{2^{d_i+2}}{d_i+2}\left(1+\varepsilon_1(\boldsymbol d)\right),\qquad \lim_{g\to\infty}\max_{\boldsymbol d\in\widehat\Pi_{4g-4}}|\varepsilon_1(\boldsymbol d)|=0. \] The error must tend to zero uniformly over the indicated partitions, rather than only for a fixed zero pattern. [S2] [S3]

Short English statement

Prove the stated product formula for the Masur--Veech volumes of all quadratic-differential strata with at most $\log g$ simple poles, with an error uniform as genus grows.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3600001 (AMR-035-0001). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Amol Aggarwal, Vincent Delecroix, Élise Goujard, Peter Zograf and Anton Zorich, Conjectural Large Genus Asymptotics of Masur–Veech Volumes and of Area Siegel–Veech Constants of Strata of Quadratic Differentials (2019), Conjecture 1, Remark 1, §3 regular-point normalization. https://arxiv.org/abs/1912.11702.
• S3 Jayadev S. Athreya, Alex Eskin and Anton Zorich, Right-angled billiards and volumes of moduli spaces of quadratic differentials on CP1 (2016), §4.1, Convention 4.1, equations (4.2)–(4.5); §4.13. https://arxiv.org/abs/1212.1660.