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.
Highlights
Let π : X → M be a holomorphic fibration with compact fibers and L a relatively ample line b√undle over X, i.e. there is a smooth metric φ on L such that the first Chern form
Soulé computed the curvature of Quillen metric for a locally Kähler family and obtained the differential form version of Grothendieck-Riemann-Roch Theorem
From Theorem 2.2 and (2.6), the curvature of L2-metric (see (2.3)) on Ek is given by c(kφ)|u|2e−kφ +
Summary
Let π : X → M be a holomorphic fibration with compact fibers and L a relatively ample line b√undle over X , i.e. there is a smooth metric (weight) φ on L such that the first Chern form. Soulé computed the curvature of Quillen metric for a locally Kähler family and obtained the differential form version of Grothendieck-Riemann-Roch Theorem. They proved that as holomorphic bundles, λy ∼=. Remark 1.4 The asymptotic of analytic torsion has been studied by [5,10] It is proved in [10, Theorem 8] the coefficients of kn, kn log k are topological invariants. 2, we fix notation and recall some basic facts on Berndtsson’s curvature formula of L2-metric, the asymptotic expansion of Bergman kernel for bundles, analytic torsion and Quillen metric. By comparing with their expansions, we prove Corollary 1.3
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