Families of projective bundles that admit coupled Kähler-Einstein metrics but no Kähler-Einstein metrics

Abstract

We construct two families of projective bundles that admit two-coupled Kähler-Einstein (cKE) metrics but do not ordinary Kähler-Einstein metrics in sufficiently high dimensions. We also prove that no such examples arise among toric Fano threefolds, for any decomposition of $c_1(X)$, when considering the natural projection onto ${\mathbb{P}}^1$.

Conjecture 1.3 has been proved