Projective bundles that admit coupled K\"ahler-Einstein metrics but no K\"ahler-Einstein metrics

TL;DR

This paper constructs explicit higher-dimensional toric Fano manifolds that admit coupled K"ahler-Einstein metrics without supporting traditional K"ahler-Einstein metrics, expanding understanding of metric existence on complex manifolds.

Contribution

It provides explicit examples of projective bundles over products of projective spaces that admit coupled KE metrics but no KE metrics, using Hultgren's polytope approach.

Findings

Explicit examples of such manifolds in higher dimensions.

Conjecture that such examples exist in all dimensions n ≥ 4.

Abstract

Using Hultgren's polytope formulation of the existence of coupled K\"ahler-Einstein (cKE) metrics on toric Fano manifolds, we construct explicit higher-dimensional toric Fano manifolds that admit two coupled K\"ahler-Einstein metrics but no ordinary K\"ahler-Einstein metrics. In particular, we exhibit such examples among certain projective bundles over products of projective spaces. Motivated by these constructions, we conjecture that examples of this type exist in all dimensions $n\geq 4$.