The asymptotics of the L²-curvature and the second variation of analytic torsion on Teichm\"uller space

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 $k\to\infty$.