Slope stability of tangent bundles of smooth toric Fano varieties

Abstract

We classify the anticanonical slope stability of tangent bundles for all $8{,}630$ smooth toric Fano varieties in dimensions three through six, by exact evaluation of Klyachko's criterion, and determine polystability (equivalently, the existence of a Hermitian--Einstein metric with respect to an anticanonical K\"ahler form) in every strictly semistable case. EveryK\"ahler--Einstein variety in the census has polystable tangent bundle, whereas the converse fails widely: $102$ of the $109$ five-folds with stable tangent bundle are not K\"ahler--Einstein. The census singles out one construction at high Picard rank, which we introduce in general: root-twisted toric $\mathrm{dP}$-fibrations over products of projective lines, parametrized by roots of $A_2$. Every nonempty multiset of nonzero root twists with vanishing sum produces a stable tangent bundle, in every dimension. Among these zero-sum twists, the resulting variety is K\"ahler--Einstein if and only if the multiset is invariant under negation or under the order-three rotation of the root hexagon. Thus stable toric Fanos of Picard rank $n+2$ exist for every $n\ge4$; vanishing twist sum does not force the K\"ahler--Einstein property, while a separate unbalanced family shows that a vanishing sum is not necessary for stability. The proof reduces the slope inequalities for the root-twist family to integrals over the $A_2$ moment hexagon. Positive layer decompositions and sharp one-dimensional convolution estimates establish stability, while a strict ordering of the hexagon's first moments at the extreme exponents yields the K\"ahler--Einstein classification.