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 $\mathbb P^n$?
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$\begingroup$ I take it $H^m$ means dimension of m-th cohomology? $\endgroup$– MattDec 2, 2010 at 21:25
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$\begingroup$ Whoops, I fixed it. $\endgroup$– James DavidoffDec 2, 2010 at 23:11
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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.
answered Dec 3, 2010 at 3:31
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5$\begingroup$ It's true in finite characteristic as well. See section 3.5 in Fulton's "Toric Varieties" book for how to compute cohomology of line bundles on toric varieties over any field. I don't think he singles out the case of the trivial line bundle; but it is easy to plug it in and see that you get 0. $\endgroup$ Apr 28, 2011 at 19:39
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$\begingroup$ 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.) $\endgroup$– BernieApr 21, 2018 at 13:45