Deligne pairing and Quillen metric

Abstract

Let X → S be a smooth projective surjective morphism of relative dimension n, where X and S are integral schemes over C. Let L → X be a relatively very ample line bundle. For every sufficiently large positive integer m, there is a canonical isomorphism of the Deligne pairing L, . . . , L → S with the determinant line bundle Det((L − O X ) ⊗(n+1) ⊗ L ⊗m ) (see [D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the knudsen-Mumford expansion, J. Differential Geom. 78 (2008) 475-496]). If we fix a hermitian structure on L and a relative Kähler form on X, then each of the line bundles Det((L − O X ) ⊗(n+1) ⊗ L ⊗m ) and L, . . . , L carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between L, . . . , L → S and Det((L−O X ) ⊗(n+1) ⊗L ⊗m ) is compatible with these hermitian structures. This holds also for the isomorphism in [Deligne pairing and determinant bundle, Electron. Res. Announc. Math. Sci. 18 (2011) 91-96] between a Deligne paring and a certain determinant line bundle.