Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics

Abstract

Let Xn → PN be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of (X, ωFS|X) restricted to the Bergman metrics is completely determined by the X-hyperdiscriminant of format (n - 1) and the Chow form of X. As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in G if and only if the hyperdiscriminant polytope dominates the Chow polytope for all maximal algebraic tori H of G.