Eskin–Zorich large-genus Siegel–Veech constant conjecture

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Let gg tend to infinity, let μ⊢2g−2\mu\vdash 2g-2 be a partition indexing a stratum of Abelian differentials, and let carea(μ)c_{\rm area}(\mu) denote its area Siegel–Veech constant. Define the error term ϵ2(μ)\epsilon_2(\mu) by the asserted asymptotic formula. Eskin–Zorich's Siegel–Veech constant conjecture. One has

carea(μ)=12+ϵ2(μ),c_{\rm area}(\mu)=\frac{1}{2}+\epsilon_2(\mu),

where

lim⁡g→∞(max⁡μ⊢2g−2∣ϵ2(μ)∣)=0.\lim_{g\to\infty}\left(\max_{\mu\vdash 2g-2}|\epsilon_2(\mu)|\right)=0.

This is the predicted uniform large-genus limit for area Siegel–Veech constants. The paper verifies it for the stratum with one zero and for the principal stratum, while the uniform assertion over all strata remains open in the source.

References

Primary source

Adrien Sauvaget, “Volumes and Siegel-Veech constants of H(2g-2) and Hodge integrals”, arXiv:1801.01744 (2018).

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