Estimates and asymptotics of Teichm\"uller modular forms

Abstract

In this article, we derive estimates of Teichm\"uller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, 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 $\pi:\mathcal{C}_{g}\longrightarrow\mathcal{M}_{g}$ be the universal surface. For any $n\geq 1$, 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 $\lambda_{n}:=\mathrm{det}(\Lambda_{n})$ denote the determinant line bundle of the vector bundle $\Lambda_{n}$, whose sections are known as Teichm\"uller modular forms. The complex vector space of Teichm\"uller modular forms is equipped with Quillen metric, which is denoted by $\|\cdot\|_{\mathrm{Qu}}$.