Characteristic-free characterizations of projective space for smooth toric varieties

We prove that a smooth projective toric variety over an algebraically closed field of arbitrary characteristic is isomorphic to a projective space if its tangent bundle contains an ample locally free subsheaf of positive rank. No torus-equivariant structure on this subsheaf is assumed. The proof uses primitive relations, the toric Euler sequence, and characteristic-free cohomological lifting. As an application of the same method, we establish the toric form of Beauville’s cohomological characterization of projective spaces and quadrics, including the polarization. Without the toric hypothesis, both characterizations fail in characteristic two.