# Inferring properties of toric manifolds through Delzant's description

Let $$(M,\omega, \mathbb{T})$$ be a symplectic toric manifold. It is well-known that the properties of $$M$$ can be retrieved by looking at the moment polytope $$\Delta$$ image of the momentum map $$\mu : M \to \text{Lie}(\mathbb{T})^*, \quad \Delta := \mu(M)$$ associated with the $$\mathbb{T}$$-action on $$M$$. If $$\Delta$$ has $$n$$ facets, it is given by $$\Delta = \lbrace x \in \text{Lie}(\mathbb{T})^* \ | \ \langle x, v_j \rangle + a_j \geq 0, \ \text{for all} \ j=1,...,n \rbrace,$$ where $$\langle.,.\rangle$$ denotes the natural pairing between $$\text{Lie}(\mathbb{T})$$ and its dual $$\text{Lie}(\mathbb{T})^*$$, the $$v_j$$'s are primitive vectors in the integer lattice $$\text{Lie}(\mathbb{T})_{\mathbb{Z}} := \exp(\text{Lie}(\mathbb{T}) \to \mathbb{T})$$, and $$a = (a_1,...,a_n) \in \mathbb{R}_{\geq 0}^n \setminus \{0\}$$. For instance, we have:

1. $$M$$ is smooth if and only if each $$k$$-codimensional face of $$\Delta$$ is the intersection of exactly $$k$$ facets;
2. $$M$$ is compact if and only if the $$k$$ conormals associated to any such $$k$$-codimensional face can be extended to an integer basis of the lattice $$\text{Lie}(\mathbb{T})_{\mathbb{Z}}$$.
3. The integral cohomology of $$M$$ can be described by means of the fan associated with the polytope $$\Delta$$. It is defined as follows: Let $$\Gamma$$ be a face of $$\Delta$$. Its associated cone is defined by $$\sigma_{\Gamma} := \underset{r > 0} \bigcup r(\Delta - x),$$ where $$x$$ is any point in the interior of $$\Gamma$$. The dual cone of $$\sigma_{\Gamma}$$ is $$\sigma_{\Gamma}^* = \lbrace v \in \text{Lie}(\mathbb{T}) \ | \ \langle x, v \rangle \geq 0, \ \text{for all} \ x \in \Gamma \rbrace.$$ The fan $$\Sigma(\Delta)$$ associated with $$\Delta$$ is the set of all dual cones $$\sigma_{\Gamma}^*$$, for $$\Gamma$$ a face of $$\Delta$$. One can then show that the cohomology of $$M$$ is given by $$H^*(M; \mathbb{Z}) = \mathbb{Z}[u_1,...,u_n] / I + J,$$ where $$I$$ and $$J$$ are ideals computed in terms of the fan $$\Sigma(\Delta)$$. I refer to the books of Audin "Torus actions on symplectic manifolds", or "Toric varieties" of Cox for more details on that.

The moment polytope $$\Delta$$ can be identified with another polytope through Delzant's construction. The standard $$(S^1)^n$$-action on $$\mathbb{C}^n$$ is induced by the momentum map. $$P : \mathbb{C}^n \to \mathbb{R}^{n*}, \quad (z_1,...,z_n) \mapsto \pi(|z_1|^2,...,|z_n|^2).$$ Endow $$\mathbb{R}^n$$ with its standard basis $$(e_1,...,e_n)$$, and consider the following surjective linear map $$\beta : \mathbb{R}^n \to \text{Lie}(\mathbb{T}), \quad e_i \mapsto v_i.$$ It induces a map $$[\beta] : \mathbb{R}^n / \mathbb{Z}^n \to \mathbb{T}.$$ Denote by $$\mathbb{K} \subset \mathbb{R}^n$$ its kernel. It has $$\text{Lie}(\mathbb{K}) = \ker \beta$$ as Lie algebra, and if $$\iota : \text{Lie}(\mathbb{K}) \hookrightarrow \mathbb{R}^n$$ denotes the inclusion, the momentum map associated with the $$\mathbb{K}$$-action on $$\mathbb{C}^n$$ is given by $$P_{\mathbb{K}} := \iota^* \circ P : \mathbb{C}^n \to \text{Lie}(\mathbb{K})^*.$$ Then $$\mathbb{K}$$ acts freely on the regular level set $$P_{\mathbb{K}}^{-1}(p), \quad p := \iota^*(a),$$ and one can show that the standard symplectic form on $$\mathbb{C}^n$$ induces a well-defined symplectic form $$\omega_p$$ on the quotient $$P_{\mathbb{K}}^{-1}(p) / \mathbb{K}$$. By Delzant's theorem, there is an equivariant symplectomorphism $$(M, \omega, \mathbb{T}) \simeq (P_{\mathbb{K}}^{-1}(p) / \mathbb{K}, \omega_p, (S^1)^n / \mathbb{K}),$$

Through this isomorphism, there is a natural identification $$\Delta \simeq (\iota^*)^{-1}(p) \cap \Pi,$$ where $$\Pi := \mathbb{R}_{\geq 0}^{n*}$$ is the first orthant. Indeed, a point $$x \in \text{Lie}(\mathbb{T})^*$$ lies in $$\Delta$$ if and only if $$\langle x, v_j \rangle + a_j \geq 0$$ for all $$j = 1,...,n$$. But $$\langle x, v_j \rangle + a_j = \langle x, \beta(e_j) \rangle + a_j = \langle \beta^*(x) + a, e_j \rangle,$$ and we have $$\text{Im} \beta + a = \ker \iota^*$$.

I am trying to understand how to interpret the compactness and smoothness of $$M$$ in terms of the polytope $$(\iota^*)^{-1}(p) \cap \Pi$$. More precisely, I would like to show the following:

1. $$M$$ is compact if and only if $$\ker \iota^* \cap \Pi = \{0\}$$;
2. $$M$$ is smooth if and only if the projections of any two $$k$$-dimensional faces of $$\Pi$$ which cover $$p$$ when projected to $$\text{Lie}(\mathbb{K})^*$$ are isomorphic over $$\mathbb{Z}$$;
3. The integral cohomology of $$M$$ is given by $$H^*(M; \mathbb{Z}) \simeq \mathbb{Z}[u_1,...,u_n] / I + J,$$ where $$I$$ is the ideal generated by polynomials which vanish on the lattice $$Lie({\mathbb{K}})_{\mathbb{Z}} := \ker (\exp: \text{Lie}(\mathbb{K}) \to \mathbb{K})$$, and $$J$$ is generated by monomials $$u_1^{m_1}...u_n^{m_n}$$ such that $$m = (m_1,...,m_n) \in \mathbb{R}_{\geq 0}^n$$, considered as a function on $$\mathbb{R}_{\geq 0}^{n*}$$, assumes strictly positive values on the (vertices) of $$(\iota^*)^{-1}(p) \cap \Pi$$.

The two first points seem rather intuitive, but I having trouble to prove them properly. As for the third one, it requires to study the relation between the fan associated with the polytope $$\Delta$$, and the faces of $$(\iota^*)^{-1}(p) \cap \Pi$$, which I don't understand.

This description appears in a paper of Alexander Givental called "A fixed point theorem for toric manifolds", but it I haven't seen it explained properly anywhere. Any help will be appreciated.

Thanks a lot!