Deligne pairing and determinant bundle

TL;DR

This paper establishes a natural isomorphism between the Deligne pairing of line bundles and the determinant line bundle in the context of smooth projective morphisms over complex schemes.

Contribution

It provides a new natural isomorphism linking Deligne pairings with determinant line bundles for line bundles over smooth projective morphisms.

Findings

Demonstrates the isomorphism explicitly in the complex algebraic setting.

Connects geometric line bundle constructions with algebraic determinant line bundles.

Enhances understanding of line bundle interactions in algebraic geometry.

Abstract

Let $X \rightarrow S$ be a smooth projective surjective morphism, where $X$ and $S$ are integral schemes over complex numbers. Let L_0, L_1, .... L_{n-1}, L_{n} be line bundles over $X$. There is a natural isomorphism of the Deligne pairing $<L_{0},...,L_{n}>$ with the determinant line bundle ${\rm Det}(\otimes_{i=0}^{n} (L_i- {\mathcal O}_{X}))$.