Conjecture on higher-dimensional toric Fano manifolds with coupled but no ordinary Kähler–Einstein metrics

The constructions concern toric Fano manifolds and coupled Kähler–Einstein metrics, meaning coupled Kähler–Einstein metrics associated with a decomposition of the first Chern class. The examples include projective bundles over products of projective spaces.

Higher-dimensional coupled Kähler–Einstein conjecture. For every integer $n\geq 4$, there exists a toric Fano manifold of dimension $n$ that admits a two-coupled Kähler–Einstein metric but does not admit an ordinary Kähler–Einstein metric. More precisely, for any integer $r\geq 2$, consider

$$X_r=\mathbb{P}_{\mathbb{C} P^r\times \mathbb{C} P^{r-1}}(\mathcal O\oplus \mathcal O(-1,1)),\qquad \dim X_r=2r,$$

and

$$Y_r=\mathbb{P}_{\mathbb{C} P^{r+1}\times \mathbb{C} P^{r-1}}(\mathcal O\oplus \mathcal O(-1,1)),\qquad \dim Y_r=2r+1.$$

These toric Fano manifolds form families of non-Kähler–Einstein toric Fano manifolds that nevertheless admit two-coupled decompositions of their first Chern classes supporting coupled Kähler–Einstein metrics.

This conjecture is motivated by the explicit five- and six-dimensional constructions in the paper and predicts that the phenomenon persists in every dimension at least four. The cited results establish the initial examples, while the general assertion remains open.