Cohomology of $\mathcal O_X$ for toric varieties

Question

Motivated by my ignorance here, if $X$ is a projective toric variety, is $$H^m(X, \mathcal O_X) \cong \begin{cases} 0 & m > 0 \\ \mathbb C & m = 1 \end{cases} $$ as for $\mathbb P^n$?

Answer 1

Yes, this is true, at least for varieties over the complex numbers $\mathbb{C}$. Indeed, a toric variety over an algebraically closed field is rational (i.e., birational to projective space). In characteristic zero, rational connectedness is a birational invariant, so toric varieties are rationally connected. Finally, any rationally connected variety is $\mathcal{O}$-acyclic, which is the name for the conclusion that you want. See e.g. here for this last implication.

The conclusion might well hold more generally; I am not an expert in these matters. You may want to ask your question on MathOverflow if you are not satisfied with this answer.