TITLE: Cohomology of $\mathcal O_X$ for toric varieties
QUESTION [5 upvotes]: 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 $\mathbb P^n$?

REPLY [10 votes]: 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.