Minimization of Arakelov K-energy for many cases

TL;DR

This paper proves that for various polarized varieties over algebraic numbers, the Arakelov K-energy is minimized in certain models, extending conjectures and analogies from complex geometry to Arakelov theory.

Contribution

It establishes the minimization of Arakelov K-energy for a broad class of polarized varieties, including K-trivial, K-ample, Fano, and minimal models, confirming conjectures in Arakelov geometry.

Findings

Minimization of Arakelov K-energy in various polarized varieties.

Extension of conjectures from complex to Arakelov geometry.

Identification of models that minimize the Arakelov K-energy.

Abstract

We prove that for various polarized varieties over $\overline{\mathbb{Q}}$, which broadly includes K-trivial case, K-ample case, Fano case, minimal models, certain classes of fibrations, certain metrized "minimal-like" models minimizes the Arakelov theoretic analogue of the Mabuchi K-energy, as conjectured in [Od15]. This is an Arakelov theoretic analogue of [H22b].