On Homothetic Balanced Metrics

TL;DR

This paper investigates the set of balanced metrics homothetic to a given one on compact complex manifolds, establishing finiteness results under certain geometric conditions related to Kähler-Einstein and constant scalar curvature metrics.

Contribution

It proves the finiteness of homothetic balanced metrics on manifolds with specific Kähler-Einstein and constant scalar curvature properties.

Findings

Finiteness of homothetic balanced metrics on manifolds with non-positive Kähler-Einstein metrics.

Finiteness results for non-homogenous toric Kähler-Einstein manifolds of dimension ≤ 4.

Finiteness for manifolds with Arezzo-Pacard constant scalar curvature metrics.

Abstract

In this paper we study the set of balanced metrics (in Donaldson's terminology) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kahler metrics. We prove that this set is finite when $M$ admits a non-positive Kahler-Einstein metric, in the case of non-homogenous toric Kaehler-Einstein manifolds of dimension $\leq 4$ and in the case of Arezzo-Pacard constant scalar curvature metrics.