Deligne Pairings and Discriminants of Algebraic Varieties

Let V be a finite dimensional complex vector space and V^* its dual and let X in P(V) be a smooth projective variety of dimension n and degree d at least two. For a generic n-tuple of hyperplanes H_1,...,H_n in P(V^*)^n, the intersection of X with H_1,...,H_n consists of d distinct points. We define the "discriminant of X", to be the the set D(X) of n-tuples for which the set-theoretic intersection is not equal to d points. Then D(X) is a hyper surface in P(V^*)^n and the set of defining polynomials, which is a one-dimensional vector space, is called the "discriminant line". We show that this line is canonically isomorphic to the Deligne pairing where K is the canonical line bundle of X and L is the restriction of the hyperplane bundle to X. As a corollary, we obtain a generalization of Paul"s formula which relates the Mabuchi K-energy on the space of Bergman metrics to \Delta(X), the "hyperdiscriminant of X".