Discriminants and Higher K-energies on Polarized K\"ahler Manifolds

TL;DR

This paper explores the relationship between higher K-energy functionals on polarized K"ahler manifolds and their expression as energies of tensor pairs, analyzing their asymptotic behavior and boundedness.

Contribution

It demonstrates that higher K-energy functionals restricted to Bergman metrics can be expressed as energies of tensor pairs, linking geometric analysis with algebraic invariants.

Findings

Higher K-energy functionals are expressible as energies of tensor pairs.

Asymptotic behavior of these functionals along 1-parameter subgroups is characterized.

Boundedness properties of the functionals are established.

Abstract

Given a compact polarized K\"ahler manifold $X\hookrightarrow\mathbb{CP}^N$, the space of Bergman metrics on $X$, parameterized by $\mathrm{SL}(N+1,\mathbb{C})$, corresponds to a dense set in the space of K\"ahler potentials in the K\"ahler class as $N\to\infty$. Critical points of the $k$th K-energy functional, which is defined on the K\"ahler class, correspond to metrics with harmonic $k$th Chern form. In this paper it is shown that the higher K-energy functionals, when restricted to the Bergman metrics, are expressible as the energies of certain pairs of vectors (tensors products of discriminants). Consequentially, we obtain results on the asymptotic behavior of these functionals along 1-parameter subgroups and their boundedness properties.