Discriminants and Higher K-energies on Polarized Kähler Manifolds

Abstract

Given a compact polarized Kähler 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 Kähler potentials in the Kähler class as $N\to \infty$. Critical points of the $k$th K-energy functional, which is defined on the Kähler 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.

15 pages

References in corpus (2)

• Stable Pairs and Coercive Estimates for The Mabuchi Functional
• A Numerical Criterion for K-Energy maps of Algebraic Manifolds