Let $X$ be a smooth projective variety over the complex numbers. Is $X$ a global complete intersection inside a smooth projective toric variety?
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1$\begingroup$ Projective $n$-space is a smooth projective toric variety, and to me "global complete intersection" is just another way to say "complete intersection" (as opposed to "local complete intersection"). So the answer is yes, unless you mean something different by "global complete intersection". $\endgroup$– Simon L Rydin MyersonCommented Mar 19, 2018 at 8:47
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2$\begingroup$ I do not understand. Not all smooth projective varieties are global complete intersections in a projective space, for instance because in higher dimension a complete intersection in $\mathbb P^n$ is necessarily simply connected. $\endgroup$– Francesco PolizziCommented Mar 19, 2018 at 9:04
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2$\begingroup$ The question becomes more interesting if you add "set-theoretic". Then I think very little is known even for curves. $\endgroup$– Piotr AchingerCommented Mar 19, 2018 at 9:43
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1$\begingroup$ Ah sorry, I somehow read "let $X$ be a smooth projective complete intersection", which made the question trivial. $\endgroup$– Simon L Rydin MyersonCommented Mar 19, 2018 at 9:53
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$\begingroup$ @PiotrAchinger Is it expected, ie. conjectured, that every smooth projective variety is a set theoretic complete intersection in a toric variety? At least I know of no example of a smooth projective variety that isn't. $\endgroup$– user120812Commented Mar 19, 2018 at 16:33
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1 Answer
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Any smooth projective toric variety is rational, in particular simply connected.
Then, by the Lefschetz hyperplane 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.
answered Mar 19, 2018 at 9:02