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- Title
Deligne pairing and Quillen metric.
- Authors
Biswas, Indranil; Schumacher, Georg
- 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 (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 and 〈L,...,L〉 carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between 〈L,...,L〉 → S and 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.
- Subjects
ISOMORPHISM (Mathematics); PROJECTIVE geometry; SURJECTIONS; DETERMINANTAL rings; HERMITIAN structures
- Publication
International Journal of Mathematics, 2014, Vol 25, Issue 14, p-1
- ISSN
0129-167X
- Publication type
Article
- DOI
10.1142/S0129167X14501225