Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics

TL;DR

This paper establishes a relationship between Mabuchi energy asymptotics and algebraic invariants like hyperdiscriminants and Chow forms for smooth algebraic varieties, linking geometric stability to polytope dominance.

Contribution

It reveals that Mabuchi energy on Bergman metrics is fully determined by hyperdiscriminants and Chow forms, and characterizes boundedness via polytope dominance for degenerations.

Findings

Mabuchi energy determined by hyperdiscriminant and Chow form

Boundedness of Mabuchi energy characterized by polytope dominance

Hyperdiscriminant polytope dominates Chow polytope iff Mabuchi energy is bounded

Abstract

Let X be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of 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 .