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