The first result provides a generalization of Deligne formulae to the case where \(n \geq 0:\)
Theorem 1. Let \(\pi : X \rightarrow B\) be a proper flat morphism of integral schemes of relative dimension \(n \geq 0\) and let \(L \rightarrow X\) be a line bundle which is very ample on the fibers.
1) There is a canonical functorial isomorphism \[ \lambda_{n+1}(L, X, B) = \langle L,..., L \rangle _{X/B}. \] 2) If \(X\) and \(B\) are smooth, and \(K\) is the relative canonical line bundle of \(X \rightarrow B,\) then there is a canonical functorial isomorphism \[ \lambda^{2}_{n}(L, X, B) = \langle L^{n}K^{-1}, L,..., L \rangle _{X/B}, \] where the right sides of 1) and 2) are Deligne pairings of \(n + 1\) line bundles.
The authors establish formulas for the first two terms in the Knudsen-Mumford expansion for \(\det(\pi_{*}L^{k})\) in terms of Deligne pairings of \(L\) and the relative canonical bundle \(K.\) As a corollary, they showed that when \(X\) is smooth, the line bundle \(\eta\) associated to \(X \rightarrow B,\) which was introduced by D.H. Phong and J. Sturm [Commun. Anal. Geom. 11, No. 3, 565–597 (2003; Zbl 1098.32012)], coincides with the CM bundle defined by S. Paul and G. Tian [CM stability and the generalized Futaki invariant, arXiv:math.AG/0605278].