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

We consider the relative canonical line bundle $K_{\mathcal{X}\mathrm{/}\mathcal{T}}$ and a relatively ample line bundle $(L,e^{-\varphi })$ 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 $\sqrt{-1}\mathrm{\partial }\bar{\mathrm{\partial }}\varphi {\mathrm{\mid }}_{{\mathcal{X}}_y}$ 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 $E^k={\pi }_{\ast }(L^k+K_{\mathcal{X}\mathrm{/}\mathcal{T}})$. As a consequence we prove that the second variation of analytic torsion ${\tau }_k(\bar{\mathrm{\partial }})$ 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 $l\ge 0$ as $k\to \mathrm{\infty }$.