Estimates and asymptotics of Teichm\"uller modular forms

TL;DR

This paper provides estimates for Teichmüller modular forms and their invariants, advancing understanding of their asymptotic behavior on moduli spaces of Riemann surfaces.

Contribution

It introduces new estimates and asymptotic bounds for Teichmüller modular forms and related invariants on moduli spaces.

Findings

Derived bounds for Teichmüller modular forms.

Established asymptotic behavior of associated invariants.

Analyzed the impact on moduli space geometry.

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 $\|\cdot\|_{\mathrm{Qu}}$.