On slope stability of tangent sheaves on smooth toric Fano varieties
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Date
30/06/2023Author
Reynolds, William
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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.