Deligne pairing and Quillen metric

TL;DR

This paper establishes a compatibility result between the Deligne pairing and determinant line bundles equipped with hermitian structures in the setting of smooth projective morphisms over complex schemes.

Contribution

It proves that the canonical isomorphism between Deligne pairing and determinant line bundles respects their hermitian structures, extending previous results to a broader context.

Findings

The isomorphism is compatible with hermitian structures.

This compatibility holds for Deligne pairings and determinant line bundles.

The result applies to a class of morphisms over complex schemes.

Abstract

Let $X\rightarrow S$ be a smooth projective surjective morphism of relative dimension $n$, where $X$ and $S$ are integral schemes over $\mathbb C$. Let $L\rightarrow X$ be a relatively very ample line bundle. For every sufficiently large positive integer $m$, there is a canonical isomorphism of the Deligne pairing $\langle L ,\cdots , L\rangle\rightarrow S$ with the determinant line bundle ${\rm Det}((L- {\mathcal O}_{X})^{\otimes (n+1)}\otimes L^{\otimes m})$ \cite{PRS}. If we fix a hermitian structure on $L$ and a relative K\"ahler form on $X$, then each of the line bundles ${\rm Det}((L- {\mathcal O}_{X})^{\otimes (n+1)}\otimes L^{\otimes m})$ and $\langle L\, ,\cdots\, ,L\rangle$ carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between $\langle L\, ,\cdots\, ,L\rangle\longrightarrow S$ and ${\rm Det}((L- {\mathcal O}_{X})^{\otimes (n+1)}\otimes L^{\otimes m})$ is compatible with these hermitian structures. This holds also for the isomorphism in \cite{BSW} between a Deligne paring and a certain determinant line bundle.