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Perhaps I might add that the "strong Kodaira vanishing" holds more generally for smooth projective toric varieties in any characteristic. This goes back to Danilov. This includes your case 1 of course. I can't remember how he did this, but an argument observed by number of people (Fujino, myself,...) is to use a what I might call a mock Frobenius splitting argument. The idea is to exploit the map $\phi$ given by multiplication by $r$ on the fan. For projective space, this is just $[x_0,\ldots, x_N]\mapsto [x_0^r,\ldots, x_N^r]$. If $r>1$ is prime to the characteristic, then $\phi^*$ can be shown to give an injection $$H^q(X,\Omega_X^p\otimes L)\hookrightarrow H^q(X,\Omega_X^p\otimes L^r)$$ So choosing $r\gg 0$, we get the desired result by Serre vanishing.