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I'm trying to understand the relationship between toric varieties and their associated polytopes. There is a very crisp result in the case of symplectic toric varieties, namely that there is a 1:1 correspondence between such objects and Delzant polytopes.

If I instead start with a smooth projective toric variety, I can again construct a polytope associated to it, but I'm not sure what sort of correspondence there should be. Another version of this question is the one in the title. I'm reading Cox Little Schenck, and though the answer is no doubt in here, I'm having trouble extracting anything precise, due to my inexperience in the area.

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  • $\begingroup$ I believe that if you start with a Delzant polytope you can always cook up a symplectic manifold, it should be explained in Guillemin's book, pag 8-9 and after $\endgroup$ Jul 7, 2023 at 17:45

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Projective toric varieties are a subset of the symplectic ones, since we can pull back the Fubini-Study form. So the answer to the question in the title is 'always', and the polytopes corresponding to projective toric varieties are a subset of the Delzant polytopes.

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