Nonsimplicial local stability chambers of toric tangent bundles

We construct smooth projective toric varieties whose tangent bundles have nonsimplicial local slope-stability chambers. For every positive integer k, a double projective-line bundle over (ℙ¹)²ᵏ has an ample real polarization at which the tangent bundle is strictly semistable. Its invariant saturated subsheaves of the same slope form the ideal lattice of the complete bipartite poset with two minimal and 2k maximal elements. The tangent bundle is an extension of a sum of 2k stable line bundles by a sum of two stable line bundles, with every component extension class nonzero. At the reference class all line factors have equal slope. Their slopes after subtracting the common mean give 2k+1 independent local polarization parameters. The local semistable set is analytically equivalent to a pointed cone with section Δ¹ × Δ²ᵏ⁻¹, times a line. The varieties are weak Fano and are not Fano. The four-dimensional example has an irrational reference polarization and attains the minimum dimension and Picard rank for a nonsimplicial chamber with a nearby stable locus.

For a smooth projective toric variety whose tangent bundle is semistable at an ample real class, its invariant saturated subsheaves of equal slope define a finite poset. We express the local semistable set by order-ideal slope inequalities restricted to the degree vectors obtained from polarizations, together with a vector-space factor. No local-freeness assumption on those subsheaves is required. We determine the ordinary local topology and localize the nearby Harder–Narasimhan filtrations to the original equality lattice. For the constructed family, the equality subbundles are relative tangent bundles of smooth projections.