  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 $G/B$ of type $A_2$ and
$B_2$.
