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We give a combinatorial criterion for the tangent bundle on a smooth toricvariety to be stable with respect to a given polarisation in terms of thecorresponding lattice polytope. Furthermore, we show that for a smooth toricsurface and a smooth toric variety of Picard rank 2, there exists an ample linebundle with respect to which the tangent bundle is stable if and only if it isan iterated blow-up of projective space.

Stability of tangent bundles on smooth toric Picard-rank-2 varieties and surfaces