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Let $X$ be a Calabi-Yau complex algebraic variety. If it is projective, we can talk about its geometric genus $p_g=h^{\dim X, 0}$, and the Calabi-Yau condition says that $p_g=1$.

Now, one might be interested in the so-called "local" Calabi-Yau varieties, which are not projective. Some of them are of greatest interest because they are toric. So my question is:

Q. Is it impossible for a smooth toric Calabi-Yau variety to be projective?

Put in another way: is the genus of a smooth projective toric variety always different from $1$?

Thank you for your time and your help!

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    $\begingroup$ Yes, it is impossible. Toric varities are rational; a smooth projective rational variety has $p_g=0$. $\endgroup$
    – user64687
    Aug 21, 2014 at 17:38
  • $\begingroup$ Yes. Thank you! $\endgroup$
    – Brenin
    Aug 22, 2014 at 14:58

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Yes, it is impossible. Toric varities are rational; a smooth projective rational variety has $p_g=0.$

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