The asymptotics of the $L^2$-curvature and the second variation of analytic torsion on Teichm\"uller space

TL;DR

This paper investigates the asymptotic behavior of the $L^2$-curvature and the second variation of analytic torsion on Teichmüller space, revealing decay rates of the torsion's second variation as the tensor power increases.

Contribution

It provides new curvature asymptotics for $L^2$-metrics and Quillen metrics on direct image bundles over Teichmüller space with constant scalar curvature metrics.

Findings

Curvature of $L^2$-metric and Quillen metric asymptotically computed.

Second variation of analytic torsion decays faster than any polynomial rate.

Results hold for high tensor powers $k$ of the line bundle.

Abstract

We consider the relative canonical line bundle $K_{\mathcal{X}/\mathcal{T}}$ and a relatively ample line bundle $(L, e^{-\phi})$ over the total space $ \mathcal{X}\to \mathcal{T}$ of fibration over the Teichm\"uller space by Riemann surfaces. We consider the case when the induced metric $\sqrt{-1}\partial\bar{\partial}\phi|_{\mathcal{X}_y}$ on $\mathcal{X}_y$ has constant scalar curvature and we obtain the curvature asymptotics of $L^2$-metric and Quillen metric of the direct image bundle $E^k=\pi_*(L^k+K_{\mathcal{X}/\mathcal{T}})$. As a consequence we prove that the second variation of analytic torsion $\tau_k(\bar{\partial})$ satisfies \begin{align*} \partial\bar{\partial}\log\tau_k(\bar{\partial})=o(k^{-l}) \end{align*} at the point $y\in\mathcal{T}$ for any $l\geq 0$ as $k\to\infty$.