Nonsimplicial local stability chambers of toric tangent bundles

We construct smooth projective toric varieties with nonsimplicial local slope-stability chambers for their tangent bundles. In every even dimension 2k+2 ≥ 4, a fibre product of two projective-line bundles over (ℙ¹)²ᵏ admits an ample real class at which the tangent bundle is strictly semistable. Near this class, the semistable locus is analytically isomorphic to a pointed cone over Δ¹ × Δ²ᵏ⁻¹, times a line, where Δʲ denotes the j-simplex. The torus-invariant saturated subsheaves of equal slope form the ideal lattice of the complete bipartite poset with two minimal and 2k maximal elements, and the stable line factors realize every centered slope direction. These varieties are weak Fano and not Fano. The fourfold has minimal dimension and Picard rank among toric varieties with such a chamber and nearby stable polarizations. We also prove a local normal form at any ample real class where the tangent bundle of a smooth projective toric variety is semistable. It retains precisely the slope directions realized by polarizations, requires no local-freeness assumption on equal-slope subsheaves, and localizes every nearby Harder–Narasimhan filtration to their equality lattice.