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

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.

• Is there no positivity condition on $A$? If not then this is just equivalent to taking the gcd of $\langle c, A \rangle$ over a basis of values of $A$. It seems easy to have this to be $1$ in the toric case, for instance for $\mathbb P^1(\mathbb C) \times \mathbb P^2(\mathbb C)$. – Will Sawin Jun 22 '19 at 20:50
• Thank you for your comment. I don't really understand what positivity means on $A$, which is a homology class. By definition, $N_M$ is positive, since it is a minimum over positive integers. Your example seems wrong: the minimal Chern number of $\mathbb{P}^n(\mathbb{C})$ is $2$, for any $n>0$. Therefore $N_{ \mathbb{P}^1(\mathbb{C}) \times \mathbb{P}^2(\mathbb{C})}$ equals $2$ as well. – BrianT Jun 22 '19 at 21:46
• The minimal Chern number of $\mathbb P^n (\mathbb C)$ is $n+1$ because the first Chern class is $n+1$ times the hyperplane class (Euler sequence). – Will Sawin Jun 23 '19 at 2:12
• This also fails in complex dimension $2$. Any exceptional $E=\mathbb{P}^{1}$ with $E^{2} = -1$, has $c_{1}.E = 1$ by the adjunction formula. So take any non-minimal toric surface, for example $\mathbb{P}^{2}$ blown up in a point. So basically almost all toric surfaces have minimal Chern number equal to 1. – Nick L Jun 23 '19 at 3:13
• Thank you Will Sawin, sorry for the mistake, of course it is $n+1$ for $\mathbb{P}^n(\mathbb{C})$... – BrianT Jun 23 '19 at 9:19

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.

• Thanks a lot for your answer. – BrianT Jun 24 '19 at 8:02
• I'm not a specialist of toric surfaces and blow ups, but is it possible that your answer holds only for non smooth toric varieties ? My question was for toric manifolds. – BrianT Jun 25 '19 at 8:45
• The blow of a smooth variety in a point is always a smooth variety (in particular it is a complex manifold). The examples I gave are smooth toric surfaces. – Nick L Jun 25 '19 at 10:44