I would like to show that the minimum number of Chern $ N_M $ of a **toric** manifold $ M $ At least $ 2 $, where

$$

N_M: = underset {l> 0} { min} lbrace existence A in H_2 (M; mathbb {Z}) : langle c, A rangle = l rbrace,

$$

$ c $ denotes the first Chern class $ (M, omega) $ (for any choice of $ omega $– compatible complex structure), and $ langle.,. rangle $ It is the natural pair between cohomology and homology groups.

I do not know how to prove this, but the following interpretation of Chern's first class could help.

Leave $ (M2d, omega, mathbb {T}) $ be a toric colic, where $ omega $ It is the symplectic form and $ mathbb {T} $ is a $ d $three-dimensional bull that acts effectively and in a Hamiltonian manner in $ (M, omega) $. Visit $ M $ as a symplectic reduction of $ mathbb {C} ^ n $ by the action of a $ k $three-dimensional subtorus $ mathbb {K} subset (S ^ 1) ^ n $ (therefore identifying $ mathbb {T} simeq (S ^ 1) ^ n / mathbb {K} $), it can be shown that there is a natural isomorphism

$$

H_2 (M; mathbb {Z}) simeq text {Lie} ( mathbb {K}) _ { mathbb {Z}},

$$

where the integral network $ text {Lie} ( mathbb {K}) _ { mathbb {Z}} $ It is the core of the exponential map. $ exp: text {Lie} ( mathbb {K}) to mathbb {K} $. For any choice of $ omega $-compatible almost complex structure in $ M $, the first class of chern $ 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 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 $ It is the inclusion of Lie algebras induced by inclusion. $ mathbb {K} subset (S ^ 1) ^ n $.

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

Any help would be appreciated. Thanks in advance.