3 edited body; edited tags

Hi. I have a question.

Definition. Delzant polytope $P$ is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex $v$, the $n$ edges containing $v$ form an element of $GL(n,\mathbb{Z})$, SL(n,\mathbb{Z})$, where$n$is a dimension of$P$. (If you wonder why this condition is called smooth, See Fulton. Introduction to toric variety chap I) My question is as follow. Can dodecahedron be the Delzant polytope? I mean, is there a symplectic toric manifold whose moment map image is combinatorially equivalent to a dodecahedron? Delzant's classfication theorem of compact symplectic toric manifold is surely very strong. But I think it is very hard to check whether the given polytope (having many faces) is of Delzant type or not. If you know any reference of give me any comment, I really appriciate for your help. Thank you. 2 edited tags; deleted 6 characters in body Hi. I have a stupid question. Definition. Delzant polytope$P$is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex$v$, the$n$edges containing$v$form an element of$GL(n,\mathbb{Z})$, where$n$is a dimension of$P$. (If you wonder why this condition is called smooth, See Fulton. Introduction to toric variety chap I) My question is as follow. Can dodecahedron be the Delzant polytope? I mean, is there a symplectic toric manifold whose moment map image is combinatorially equivalent to a dodecahedron? Delzant's classfication theorem of compact symplectic toric manifold is surely very strong. But I think it is very hard to check whether the given polytope (having many faces) is of Delzant type or not. If you know any reference of give me any comment, I really appriciate for your help. Thank you. 1 # About a Delzant polytope. (In particular dodecahedron) Hi. I have a stupid question. Definition. Delzant polytope$P$is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex$v$, the$n$edges containing$v$form an element of$GL(n,\mathbb{Z})$, where$n$is a dimension of$P\$.

(If you wonder why this condition is called smooth, See Fulton. Introduction to toric variety chap I)

My question is as follow.

Can dodecahedron be the Delzant polytope? I mean, is there a symplectic toric manifold whose moment map image is combinatorially equivalent to a dodecahedron?

Delzant's classfication theorem of compact symplectic toric manifold is surely very strong. But I think it is very hard to check whether the given polytope (having many faces) is of Delzant type or not. If you know any reference of give me any comment, I really appriciate for your help.

Thank you.