Weight Polytopes and Energy Functionals of Toric Varieties

Abstract

We prove that the weight polytope of the Hurwitz form of a polarized smooth toric variety coincides with the convex hull of the characteristic vectors introduced in Ogusu and Sano (Characteristic vectors for the Hurwitz polytopes of toric varieties, preprint 2023) with respect to all regular triangulations of the momentum polytope. Our proof relies on the combination of the two slope formulas of K-energy (Boucksom et al. in J Eur Math Soc 21(9):2905–2944, 2019; Paul in Ann Math 175(1):255–296, 2012) in the toric setting.

Funding

Japan Society for the Promotion of Science(22K03325)

Fukuoka University(225001-000)