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 \(\pi _*(L^k\otimes K_{\mathcal {X}/M})\) 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.

令\（\ pi：\ mathcal {X} \ rightarrow M \）是具有致密纤维的L型同质纤维，L是\（\ mathcal {X} \）上相对较丰富的线束。我们在直接图像束\（\ pi _ *（L ^ k \ otimes K _ {\ mathcal {X} / M}）\）上获得\（L ^ 2 \） -metric和Qullien metric的曲率渐近线对于较大的k，最多可达到\（k ^ {n-1} \）的低阶项。作为应用程序，我们证明了解析扭转\（\ tau _k（\ bar {\ partial}）\）满足\（\ partial \ bar {\ partial} \ log（\ tau _k（\ bar {\ partial}） ^ 2 = o（k ^ {n-1}）\），其中n是纤维的尺寸。