On slope stability of tangent sheaves on smooth toric Fano varieties

Abstract

We prove that number 82 in the Batyrev-Sato classi cation of smooth toric Fano 4-folds is an explicit counterexample of Picard rank 4 to a long-standing conjecture due to Peternell - the fi rst known counterexample of Picard rank greater than 1. Speci cally, we use toric methods to show that the tangent sheaf admits exactly one equivariant destabilizing subsheaf and that this subsheaf is not the relative tangent sheaf of any (not necessarily elementary) Mori bration. We give the classification of the 124 smooth toric Fano 4-folds according to stability of their tangent bundles, provide the Harder-Narasimhan filtrations in all 74 cases in which the tangent bundle is unstable, and show that number 82 is the only counterexample to Peternell's conjecture among the smooth toric Fano 4-folds.

There are 3 other cases in which none of the terms of the Harder-Narasimhan fi ltration come from a Mori bration, 2 of which are blow-downs of number 82, and we examine these 3 in some detail. We fi nish with some observations and conjectures to do with stability of the tangent bundle on Fano Bott towers, and an appendix containing Macaulay2 code which can be used to compute stability and Harder-Narasimhan fi ltrations of tangent sheaves on toric varieties.