Tilting bundles via the Frobenius morphism

We show how to construct tilting bundles for a class of smooth projective varieties using characteristic $p$ methods. Given such a variety $X$, reduce it modulo a prime number and consider the direct image of the structure sheaf under the Frobenius morphism. We prove that under suitable restrictions on the characteristic, these direct images are tilting bundles for some toric Fano varieties, Del Pezzo surfaces, and flag varieties $G\mathrm{/}B$ of type $A_2$ and $B_2$.