Conjectural Large Genus Asymptotics of Area Siegel–Veech Constants

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. For a connected component $\mathcal C$ of the unit-area slice, its area Siegel--Veech constant $c_{\mathrm{area}}(\mathcal C)$ counts maximal flat cylinders with weight equal to cylinder area divided by total surface area. More explicitly, for almost every surface $S\in\mathcal C$, of area $A$, put \[ N_{\mathrm{area}}(S,L)=\sum_{\substack{Z\text{ maximal cylinder on }S\\\text{circumference}(Z)\le L}} \frac{\operatorname{Area}(Z)}{A},\qquad N_{\mathrm{area}}(S,L)\sim c_{\mathrm{area}}(\mathcal C)\,\frac{\pi L^2}{A}. \] Almost every refers to the normalized Masur--Veech measure on the component. A non-hyperelliptic component is a connected component other than a hyperelliptic component in the stratum classification.

Statement recorded in the supplied source.

For $g\ge6$, prove the uniform asymptotic \[ c_{\mathrm{area}}(\mathcal C)=\frac14\left(1+\varepsilon_2(\boldsymbol d,\mathcal C)\right),\qquad \lim_{g\to\infty}\max_{\substack{\boldsymbol d\in\widehat\Pi_{4g-4}\\\mathcal C\text{ non-hyperelliptic component of }\mathcal Q_g(\boldsymbol d)}} |\varepsilon_2(\boldsymbol d,\mathcal C)|=0. \] The component argument in the error makes explicit that the assertion concerns every indicated component. [S2] [S3]

Short English statement

Show that the area-weighted cylinder-counting constant approaches $1/4$ uniformly over all non-hyperelliptic components in the specified quadratic strata.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3600002 (AMR-035-0002). 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 2 and Remark 1. 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.