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!


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