Deligne pairing and determinant bundle

Abstract

Let $X\to S$ 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 $\mathrm{Det}({\otimes }_{i=0}^n(L_i-{\mathcal{O}}_X))$.

ERA Mathematical Sciences (to appear)