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Tippi
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There are well known examples of smooth (hence simplicial) complete toric varieties which are not projective,. See for example in Chapter 3p. 71 of Fulton's book Introduction to Toric Varieties. Any such variety gives a counterexample.

There are well known examples of smooth (hence simplicial) toric varieties which are not projective, for example in Chapter 3 of Fulton's book. Any such variety gives a counterexample.

There are well known examples of smooth (hence simplicial) complete toric varieties which are not projective. See for example p. 71 of Fulton's book Introduction to Toric Varieties. Any such variety gives a counterexample.

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Tippi
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There are well known examples of smooth (hence simplicial) toric varieties which are not projective, for example in Chapter 3 of Fulton's book. Any such variety gives a counterexample.