Deligne pairing for equidimensional morphisms

TL;DR

This paper constructs a symmetric multi-additive functor related to Deligne pairings for equidimensional morphisms, enabling the development of arithmetic intersection theory for hermitian line bundles in algebraic geometry.

Contribution

It introduces a new functorial construction for Deligne pairings in the context of equidimensional morphisms, expanding the tools available for arithmetic intersection theory.

Findings

Constructed a symmetric multi-additive functor for Deligne pairing.

Proved functorial properties of the constructed functor.

Application to defining arithmetic intersection theory for hermitian line bundles.

Abstract

Let $S$ be a noetherian normal scheme, and let $X\to S$ be a surjective projective morphism of pure relative dimension $d$. We construct a symmetric multi-additive functor $\mathcal{P}\mathrm{ic}(X)^{d+1} \to \mathcal{P}\mathrm{ic}(S)$, and prove its functorial properties. Our construction uses Elkik's and Garc\'ia's ideas, as well as algebraic Hartogs' theorem. Moreover, our results can be used to define arithmetic intersection theory of hermitian line bundles for equidimensional morphisms.