The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle

Abstract: Let $\pi:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and $L$ a relatively ample line bundle over $\mathcal{X}$. We obtain the asymptotic of the curvature of $L^2$-metric and Qullien metric on the direct image bundle $\pi_*(L^k\otimes K_{\mathcal{X}/M})$ up to the lower order terms than $k^{n-1}$ for large $k$. As an application we prove that the analytic torsion $\tau_k(\bar{\partial})$ satisfies $\partial\bar{\partial}\log(\tau_k(\bar{\partial}))^2=o(k^{n-1})$, where $n$ is the dimension of fibers.