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

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