Delzant theorem for polyhedra

Question

Delzant theorem says that there is a 1-1 correspondence between compact toric symplectic manifolds (modulo equivariant symplectomorphism) and the Delzant polytopes (modulo lattice isomorphism). The polytope is given by the image of a moment map of the torus action.

My question is

Given a Delzant polyhedron(may not be bounded), is there a unique toric symplectic manifold corresponding to the polyhedron?

Following the book "Lectures on Symplectic Geometry" by Ana Cannas da Silva (with different sign convention), the construction of a toric symplectic manifold from a Delzant polytope $P$ is as follows. Let $v_1, \dots, v_m$ denote inward primitive vectors of facets of $P$. The polypote $P$ is written as $\{x \in (\mathbb{R}^n)^\vee\mid \langle x, v_i \rangle \geq -\lambda_i\}$. Define $A\colon \mathbb{R}^m \rightarrow \mathbb{R}^n$ by $A(e_i) = v_i$. We have an exact sequence

$$0 \rightarrow K \rightarrow^{j} \mathbb{R}^m \rightarrow^{A} \mathbb{R}^n \rightarrow 0.$$ Dually, $$ 0 \rightarrow (\mathbb{R}^n)^\vee \rightarrow^{A^t} (\mathbb{R}^m)^\vee \rightarrow^{j^*} K^\vee \rightarrow 0.$$

Let $\Phi\colon \mathbb{C}^m \rightarrow (\mathbb{R}^m)^\vee$ be a moment map given by $\Phi(z) = (\pi|z_1|^2 - \lambda_1, \dots, \pi|z_m|^2 - \lambda_m)$. Now $Q:=Ker(\mathbb{R}^m/\mathbb{Z}^m \rightarrow^A \mathbb{R}^n/\mathbb{Z}^n)$ acts on $\mathbb{C}^m$ with moment map $j^* \circ \Phi$. The symplectic reduction $M$ at $0$ has residual $T^m/Q$ action whose moment map image is $P$.

It seems to me that we didn't use the fact that $P$ is bounded, so a toric manifold $M$ can be constructed from a polyhedron $P$. Do we need to assume $P$ to be bounded to say such $M$ is unique?

Answer 1

Maybe the condition of being Delzant (I'd say its enough with the "simple" condition) implies that the polytope is bounded. Do you have an example of Delzant polytope which is not bounded?

Answer 2

Yes.

However, not every noncompact toric variety can be classified in this way. More details on this classification may be found in the paper Non-Compact Symplectic Toric Manifolds by Karshon and Lerman:
https://doi.org/10.48550/arXiv.0907.2891. Theorem 6.7 shoud be especially of interest with regard to your question.

Also, you are correct that Delzant's construction (which is presented in the book by Cannas da Silva) does not rely on boundedness of the polyhedron. When you do the construction for unbounded polyhedra, however, one additional restriction must be imposed: the polyhedron must have at least one vertex.

If you have a rational, simple and smooth polyhedron with at least one vertex, Delzant's construction will give you a toric symplectic manifold that is compact if and only if the polyhedron is bounded.