Deligne pairing and determinant bundle

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 $$ with the determinant line bundle ${\rm Det}(\otimes_{i=0}^{n} (L_i- {\mathcal O}_{X}))$.


Introduction
Let X −→ S be a smooth family of complex projective curves parameterized by an integral scheme S/C. Let L 0 and L 1 be line bundles over X. In [2], P. Deligne associated to this data a line bundle L 0 , L 1 over the parameter space S. This construction is now known as the Deligne pairing. S. Zhang extended the Deligne pairing to arbitrary relative dimension [13]. Let S and X be integral schemes over C, and let (1.1) f : X −→ S be a smooth projective surjective morphism. Let n be the dimension of the fibers of f . Take algebraic line bundles L 0 , L 1 , · · · , L n−1 , L n over X. The Deligne pairing, [13], is a line bundle L 0 , · · · , L n −→ S (the construction is briefly recalled in Section 2). The map Pic(X) n+1 −→ Pic(S) defined by (L 0 , · · · , L n ) −→ L 0 , · · · , L n is symmetric, and it is bilinear with respect to the group structure defined by the tensor product of line bundles and dualization; it is also compatible with base change.
Given a locally free coherent sheaf F on X, we have a line bundle Det(F ) on S (see [7], [1]). This extends to a homomorphism to Pic(S) from the Grothendieck group of locally free coherent sheaves on X.
The aim of this note is to announce the following: Theorem. There is a canonical isomorphism L 0 , · · · , L n −→ Det(⊗ n i=0 (L i − O X )).
If each L i , i ∈ [0 , n], is equipped with a C ∞ hermitian structure h i , then L 0 , · · · , L n inherits a hermitian structure [2], [13]. On the other hand, the hermitian structures h 1 , · · · , h n , the trivial hermitian structure on the trivial line bundle O X , and a relative Kähler structure on X together define a hermitian structure on the determinant bundle Det(⊗ n i=0 (L i − O X )) according to [1], [12]. The curvatures of L 0 , · · · , L n and Det(⊗ n i=0 (L i − O X )) coincide (see Proposition 5). Finally we observe that the Weil-Petersson metric for families of canonically polarized varieties can be interpreted as the curvature form of a certain Deligne pairing.
After an initial version was written by the first two authors, the referee pointed out that this work was also done by the third author. We thank the referee for this.

A canonical isomorphism
We continue with the notation of the introduction. Let L 0 , L 1 , · · · , L n−1 , L n be line bundles over X. A local trivialization of L 0 , · · · , L n over some Zariski open subset U of S is given by fixing a rational section is denoted by l 0 , . . . , l n . To describe the line bundle L 0 , · · · , L n in terms of these trivializations, we need to give the transition functions for ordered pairs of such trivializations. Let g be a rational function on X, and i ∈ [1 , n]. Assume that j =i div(l j ) = k n k Y k is finite over S, and that it has empty intersection with div(g) over an open subset U ′ ⊂ U (this subset is the intersection of two open subsets of the above type). Then l 0 , . . . , g l i , . . . , l n is another generator. The transition function is given by the following equation: It is sufficient to describe this type of transition functions, because a general transition function is a product of such functions.
Given a coherent sheaf V on S, we have a line bundle det(V ) on S (see [8, Ch. V, § 6]). For a coherent sheaf F on X, we have a line bundle [1]. For coherent sheaf F 1 and F 2 on X, define Theorem 1. Let L 0 , L 1 , · · · , L n be line bundles over X. Then there is a canonical isomorphism ϕ : We begin with the following observation: Let D ′ and D ′′ be effective divisors on X such that the intersection Y := D ′ D ′′ does not contain any divisor. Then the following equality of elements of K(X) holds: Proof. This follows immediately from the identity We will describe the determinant line bundle Det(⊗ n i=0 (L i − O X )) in terms of local trivializations and transition functions. For that purpose, we construct a covering of the base S by Zariski open subsets S ′ over which each line bundle L i , i ∈ [0 , n], is given by a divisor D + i − D − i , where both D + i and D − i are effective, such that the following conditions hold: any intersection of n hypersurfaces in the union of all these divisors is reduced, the intersection is of the expected codimension, and it is finite over the base.
We note that if Z is an intersection of n hypersurfaces in the union these divisors, then f (Z) is contained in a divisor on S because Z is of expected codimension. This choice of divisors for the line bundles gives a trivialization of Det(⊗ n i=0 (L i − O X )) over the complement of the union of all f (Z I ), where I runs over the set of n hypersurfaces in the union the divisors. This open subset, which will be denoted by S 0 , is nonempty because each f (Z I ) is of codimension at least one. There is a trivialization We now pick a rational function on X. We assume that the divisor (g) := div(g) can be included in the above system of divisors so that the above properties continue to hold for the enlarged system. The line bundle L 0 is given by the divisor  Lemma 3. The transition function t in (2.7) has the following expression: where Y σ and n σ are defined above, and g is the function in (2.5).
Theorem 1 is proved using Lemma 3. The details will appear elsewhere.
If F is a vector bundle on X equipped with a hermitian structure h F , then there is a natural hermitian structure on the line bundle Det(F ) −→ S which is constructed using h F and ω X/S [12], [1]; it is known as the Quillen metric. If F 1 and F 2 are vector bundles equipped with hermitian structure, then the hermitian structures on Det(F 1 ) and Det(F 2 ) * together induce a hermitian structure on Det( For each l ∈ [0 , n], fix a hermitian metric h j on the line bundle L j over X. These h j produce a hermitian metric on L 0 , · · · , L n [2], [13, § 1.2]. Therefore, both the line bundles Det(⊗ n i=0 (L i − O X )) and L 0 , · · · , L n are equipped with a hermitian metric.
Proposition 5. The curvature of the hermitian metric on L 0 , · · · , L n coincides with the curvature of the Quillen metric on Det(⊗ n i=0 (L i − O X )).
Proof. The Chern form of the metric on L 0 , . . . , L n equals the fiber integral (see [13]). On the other hand, a theorem of Bismut, Gillet and Soulé [1] says that the Chern form of the determinant line bundle is the degree two component of the Riemann-Roch fiber integral (3.2) , where h Q is the Quillen metric on Det(⊗ n i=0 (L i − O X )) (this theorem of [1] was extended to (smooth) Kähler fibrations over singular base spaces in [6, §12]).
Note that ch(L − O X ) = c 1 (L, h) + higher order terms .
Hence the only contribution of td(X/S) in (3.2) is the constant one, and also the higher order terms in ch(L−O X ) do not contribute. Consequently, (3.2) coincides with (3.1).
Let X −→ S be a projective family of canonically polarized varieties. Equip the relative canonical bundle K X/S with the hermitian metric that is induced by the fiberwise Kähler-Einstein metrics. It was shown in [6] that the generalized Weil-Petersson form is equal, up to a numerical factor, to the fiber integral ω W P ≃ X/S c 1 (K X/S , h) n+1 . Therefore, we have the following: Proposition 6. Let X −→ S be a projective family of canonically polarized varieties. The curvature of the metric on the Deligne pairing K X/S , . . . , K X/S given by the fiberwise Kähler-Einstein metric coincides with the generalized Weil-Petersson form ω W P on S.