Higher Energies in Kahler Geometry I

TL;DR

This paper introduces new energy functionals in Kähler geometry linked to algebraic group actions, generalizing known energies like Aubin and Mabuchi energies, and explores their asymptotic behavior and relations.

Contribution

It defines a family of energies $F_{ ext{om},l}( ho)$ associated with algebraic one-parameter subgroups, extending classical energies in Kähler geometry and connecting them to algebraic invariants.

Findings

$F_{ ext{om},l}( ho)$ reduces to Aubin energy for $l=0$.

$F_{ ext{om},l}( ho)$ coincides with Mabuchi's K-energy for $l=1$.

For $l eq 0,1$, $F_{ ext{om},l}( ho)$ aligns with Chen and Tian's functional $E_{ ext{om},l-1}( ho)$.

Abstract

Let $X\hookrightarrow \cpn $ be a smooth complex projective variety of dimension $n$. Let $\lambda$ be an algebraic one parameter subgroup of $G:=\gc$. Let $ 0\leq l\leq n+1$. We associate to the coefficients $F_{l}(\lambda)$ of the normalized weight of $\lambda$ on the $mth$ Hilbert point of $X$ new energies $F_{\om,l}(\vp)$. The (logarithmic) asymptotics of $F_{\om,l}(\vp)$ along the potential deduced from $\lambda$ is the weight $F_{l}(\lambda)$. $F_{\om,l}(\vp)$ reduces to the Aubin energy when $l=0$ and the K-Energy map of Mabuchi when $l=1$. When $l\geq 2$ $F_{\om,l}(\vp)$ coincides (modulo lower order terms) with the functional $E_{\om,l-1}(\vp)$ introduced by X.X. Chen and G.Tian.