# Cohomology of $\mathcal O_X$ for toric varieties

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 $\mathbb P^n$?

• I take it $H^m$ means dimension of m-th cohomology?
– Matt
Dec 2, 2010 at 21:25
• Whoops, I fixed it. Dec 2, 2010 at 23:11

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.
• Alternatively, one uses that the Hodge numbers $h^{p,0}$ are birational invariants. (I hope this comment might be useful to someone some day. Your answer definitely was useful for me.) Apr 21, 2018 at 13:45