Slope stability of tangent bundles of smooth toric Fano varieties

Abstract

We construct smooth toric Fano $n$-folds of Picard number $n+2$ whose tangent bundles are slope-stable with respect to the anticanonical polarization, for every $n\ge4$. The construction gives toric fibrations with fibre the del Pezzo surface of degree six over products of projective lines, parametrized by multisets of roots of $A_2$. Every nonempty multiset of nonzero roots with vanishing sum yields a stable tangent bundle. Among these multisets, the resulting variety is Kähler-Einstein if and only if the multiset is invariant under negation or under the order-three rotation of the root hexagon. We also construct stable, non-Kähler-Einstein examples with nonzero twist sum in every even dimension at least four. The proofs reduce slope inequalities and barycenter computations to integrals over the moment hexagon. For a smooth toric Fano variety with strictly semistable tangent bundle, we prove that polystability is equivalent to decomposition as a nontrivial product of smooth toric Fano varieties with anticanonically stable tangent bundles. Exact evaluation of Klyachko's criterion extends the stability classification to dimensions five and six, recovering the classifications of Steffens and Reynolds in dimensions three and four. Together with the product criterion, this determines polystability for all $8{,}630$ varieties in dimensions three through six, and hence the existence of Hermitian-Einstein metrics on their tangent bundles with respect to an anticanonical Kähler form.