Let $X$ be a smooth projective variety over the complex numbers. Is $X$ a *global complete intersection* inside a smooth projective toric variety?



Complete intersections in toric varieties
Any smooth projective toric variety is rational, in particular simply connected. 


Then, by the [Lefschetz hyperplane theorem](https://www.encyclopediaofmath.org/index.php/Lefschetz_theorem) for global complete intersections, if $\dim X \geq 3$ is a smooth complete intersection into a smooth toric variety then $\pi\_1(X)=\{1\}$.


In particular, for instance, abelian varieties of dimension at least $3$ cannot be realized as global complete intersections into a smooth toric ambient. 


