Is the minimal Chern number of a toric manifold at least 2?

Question

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.

Answer 1

Let $X$ be the blow up of $\mathbb{P}^{2}$ in a torus invariant point. The blow up is toric, in fact its moment polytope can be calculated by "cutting away" a triangular region around the vertex corresponding to the fixed point (texts on toric geometry will cover this, for example the book of Cox).

As with all blow ups (in dimension $2$), the exceptional divisor $E$ is isomorphic to $\mathbb{P}^{1}$ with self intersection number $E \cdot E = -1$. For a complex algebraic manifold $X$ we have that the first chern class of the tangent bundle is Poincaré dual to the anticanonical divisor $-K_{X}$. Now lets apply the adjunction formula to $E$.

$$K_{E} = (K_{X} + E)|_{E} $$

We know $$\int_{E} K_{E} = -2, $$ since it is topologically a $2$-sphere. Hence $$\int_{E} c_{1}(X) = \int_{E} -K_{X}|_{E} = \int_{E}(-K_{E} + E)_{E} = 2 -1 = 1.$$

Hence the minimal Chern number is $1$. Given any toric surface we can blow up a torus invariant point and the same calculation will give that the minimal chern number is $1$.

So the only remaining cases to check in dimension 2 are minimal models. Since they are rational there is only $\mathbb{P}^{2}$ and the Hirzebruch surfaces $\mathbb{F}_{i}$. The minimal Chern number of $\mathbb{P}^{2}$ is 3.the minimal Chern number of $\mathbb{P}^{1} \times \mathbb{P}^{1} = \mathbb{F}_{0}$ is 2. The minimal Chern number of $\mathbb{F}_{1}$ is $1$. In general, the minimal Chern number of $\mathbb{F}_{n}$ will be $2$ if $n$ is even and $1$ otherwise.