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

Abstract

Let pi :mathcal {X}rightarrow 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^kotimes 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.