I would like to show that the minimal Chern number $N_M$ of a **toric** manifold $M$ is at least $2$, where
$$
N_M := \underset{l>0}{\min} \lbrace \exists \ A \in H_2(M;\mathbb{Z}) \ : \ \langle c, A \rangle = l \rbrace,
$$
$c$ denotes the first Chern class of $(M,\omega)$ (for any choice of $\omega$-compatible complex structure), and $\langle.,.\rangle$ is the natural pairing between cohomology and homology groups.

I don't know how to prove this, but the following interpretation of the first Chern class might help.

Let $(M^{2d},\omega, \mathbb{T})$ be a toric manifold, where $\omega$ is the symplectic form and $\mathbb{T}$ is a $d$-dimensional torus acting effectively and in a Hamiltonian way on $(M,\omega)$. Viewing $M$ as a symplectic reduction of $\mathbb{C}^n$ by the action of a $k$-dimensional subtorus $\mathbb{K} \subset (S^1)^n$ (hence identifying $\mathbb{T} \simeq (S^1)^n / \mathbb{K}$), one can show that there is a natural isomorphism $$ H_2(M;\mathbb{Z}) \simeq \text{Lie}(\mathbb{K})_{\mathbb{Z}}, $$ where the integral lattice $\text{Lie}(\mathbb{K})_{\mathbb{Z}}$ is the kernel of the exponential map $\exp: \text{Lie}(\mathbb{K}) \to \mathbb{K}$. For any choice of $\omega$-compatible almost complex structure on $M$, the first Chern class $c \in H^2(M;\mathbb{Z}) \simeq \text{Lie}(\mathbb{K})_{\mathbb{Z}}^*$ writes: $$ c(m) = \underset{j=1}{\overset{n} \sum} m_j \quad m \in \text{Lie}(\mathbb{K})_{\mathbb{Z}} \quad \iota(m) = (m_1,...,m_n), $$ where $\iota : \text{Lie}(\mathbb{K}) \hookrightarrow \mathbb{R}^n$ is the inclusion of Lie algebras induced by the inclusion $\mathbb{K} \subset (S^1)^n$.

Of course, in general (when $M$ is not toric), $N_M$ can be equal to $1$, and one can even have that $\langle c, H_2(M;\mathbb{Z}) = 0$ (in which case one often writes $N_M = \infty$). However, since any toric manifold has a decomposition in complex cells, it seems that $N_M$ should be at least $2$.

Any help will be appreciated. Thanks in advance.

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