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

Journal article, 2021

Let π: X→ M be a holomorphic fibration with compact fibers and L a relatively ample line bundle over X. We obtain the asymptotic of the curvature of L -metric and Qullien metric on the direct image bundle π (L ⊗ K ) up to the lower order terms than k , for large k. As an application we prove that the analytic torsion τ (∂¯) satisfies ∂∂¯log(τk(∂¯))2=o(kn-1), where n is the dimension of fibers.