Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials.

*(English)*Zbl 1019.32014Let \(\Sigma \) be a compact Riemann surface of genus \(g>1\) and let \(\mu\) be a partition of \(2g-2\) into \(l=l(\mu)\) parts. Let \({\mathcal H}(\mu)\) denote the moduli space of \((l+2)\)-tuples \((\Sigma, \omega,p_1,\cdots,p_l)\) where \(\omega\) is a holomorphic differential on \(\Sigma\) with divisor \((w)=\sum \mu_i[p_i].\)

The authors consider the subspace \({\mathcal H}_1(\mu)\) of \({\mathcal H}(\mu)\) defined by the equation Area\(_{\omega}(\Sigma)=1\) (where the area is taken with respect to the metric defined by \(\omega\)), together with a measure \(\nu\) on \({\mathcal H}_1(\mu)\) defined as follows. For a set \(E \subset {\mathcal H}_1(\mu)\) that lies in the domain of a coordinate chart \(\Phi,\) let \(C\Phi(E)\) be the cone over \(\Phi(E)\) with vertex at the origin, and set \(\nu(E)\colon =\text{vol}(C\Phi(E)).\) This measure is invariant under the action of \(SL(2,\mathbb R)\) on \({\mathcal H}_1(\mu)\) and it was known by results of Masur and Veech that \(\nu({\mathcal H}_1(\mu))<\infty.\) The authors compute \(\nu({\mathcal H}_1(\mu))\) showing in particular the rationality of \(\pi^{-2g}\nu({\mathcal H}_1(\mu)),\) that had been conjectured by Kontsevich and Zorich.

The authors consider the subspace \({\mathcal H}_1(\mu)\) of \({\mathcal H}(\mu)\) defined by the equation Area\(_{\omega}(\Sigma)=1\) (where the area is taken with respect to the metric defined by \(\omega\)), together with a measure \(\nu\) on \({\mathcal H}_1(\mu)\) defined as follows. For a set \(E \subset {\mathcal H}_1(\mu)\) that lies in the domain of a coordinate chart \(\Phi,\) let \(C\Phi(E)\) be the cone over \(\Phi(E)\) with vertex at the origin, and set \(\nu(E)\colon =\text{vol}(C\Phi(E)).\) This measure is invariant under the action of \(SL(2,\mathbb R)\) on \({\mathcal H}_1(\mu)\) and it was known by results of Masur and Veech that \(\nu({\mathcal H}_1(\mu))<\infty.\) The authors compute \(\nu({\mathcal H}_1(\mu))\) showing in particular the rationality of \(\pi^{-2g}\nu({\mathcal H}_1(\mu)),\) that had been conjectured by Kontsevich and Zorich.

Reviewer: Elizabeth Gasparim (Las Cruces)