Vanishing theorems on toric varieties in positive characteristic

We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata–Viehweg vanishing theorem for log pairs on toric varieties in positive characteristic. These results generalize those results of Danilov, Buch–Thomsen–Lauritzen–Mehta, Musta?ǎ and Fujino to the case where concerned Weil divisors are not necessarily torus invariant.