Orbifold K\"ahler-Einstein metrics on projective toric varieties

TL;DR

This paper proves that all $Q$-factorial normal projective toric varieties admit orbifold K"ahler-Einstein metrics and characterizes their K-stability using algebraic invariants.

Contribution

It establishes the existence of orbifold K"ahler-Einstein metrics on all $Q$-factorial projective toric varieties and links K-stability to the log Cox ring and orbifold cover.

Findings

All $Q$-factorial normal projective toric varieties admit orbifold K"ahler-Einstein metrics.

K-stability of $Q$-factorial toric pairs of Picard number one is characterized algebraically.

The log Cox ring and universal orbifold cover determine K-stability in this setting.

Abstract

In this short note, we investigate the existence of orbifold K\"ahler-Einstein metrics on toric varieties. In particular, we show that every $\mathbb{Q}$-factorial normal projective toric variety allows an orbifold K\"ahler-Einstein metric. Moreover, we characterize the K-stability of $\mathbb{Q}$-factorial toric pairs of Picard number one in terms of the log Cox ring and the universal orbifold cover.