Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics

Abstract

Let $X^n\rightarrow \mathbb{P}^N$ be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of $(X,{\omega_{FS}}|_X)$ 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 $G$.