Abstract

This will be a continuation of the talk by Xuwen Zhu on our joint work concerning the regularity of the fibre hyperbolic metrics up to the singular fibres for Lefschetz fibrations. In particular this applies to the universal curve over moduli space. I will discuss the marked case with the moduli space $mathcal{M}_{g,n}$ of surfaces of genus $g$ with $n$ ordered distinct points in the stable range, $2g+nge3.$ As in the unmarked case the description of the regularity of the fibre hyperbolic metrics, up to the divisors forming the `boundary’ of the Knudsen-Deligne-Mumford compactification, implies boundary regularity for the Weil-Peterssen metric. In this case it also leads to an asymptotic description of the Takhtajan-Zograf metric which contributes to the Chern form of the determinant bundle for $arpa$ on the fibres of the universal curve.