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Jul 24, 2015 at 12:04 comment added Lazzaro Campeotti One nice variant of the argument you talk about in the last paragraph works for toric varieties in characteristic zero. They have something analogous to a Frobenius (indeed, a whole family of Frobenii) for which the map $O_X \rightarrow F_* O_X$ does indeed split.
Jul 24, 2015 at 11:42 history edited Donu Arapura CC BY-SA 3.0
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Jul 23, 2015 at 21:15 history edited Donu Arapura CC BY-SA 3.0
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Jul 23, 2015 at 21:09 history answered Donu Arapura CC BY-SA 3.0