Estimates and asymptotics of Teichmüller modular forms

Abstract

In this article, we derive estimates of Teichmüller modular forms, and associated invariants. Let ${\mathcal{M}}_g$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\ge 2$, and let ${\overline{M}}_g$ be the Deligne-Mumford compactification of ${\mathcal{M}}_g$, and we denote its boundary by $\partial {\mathcal{M}}_g$. Let $\pi :{\mathcal{C}}_g⟶{\mathcal{M}}_g$ be the universal surface. For any $n\ge 1$, let ${\Lambda }_n:={\pi }_{\ast }(T_v{\mathcal{C}}_g{)}^n$, where $T_v{\mathcal{C}}_g$ denotes the vertical holomorphic tangent bundle of the fibration $\pi$, and the fiber of ${\Lambda }_n$ over any $X\in {\mathcal{M}}_g$ is equal to $H^0(X,{\Omega }_X^{\otimes n})$, the space of holomorphic differentials of degree-$n$, defined over the Riemann surface $X$. Let ${\lambda }_n:=\det ({\Lambda }_n)$ denote the determinant line bundle of the vector bundle ${\Lambda }_n$, whose sections are known as Teichmüller modular forms. The complex vector space of Teichmüller modular forms is equipped with Quillen metric, which is denoted by $\Vert \cdot {\Vert }_{\mathrm{Qu}}$.

First version, and looking forward to comments