In an article about toric manifolds, I have seen the following notions, which I don't understand. We view a symplectic toric manifold $(M,\omega)$ as a Kähler manifold with Kähler form $\omega$, and first Chern class $c$.

It is said that the cohomology class $[\omega]$ and the first Chern class $c$ are *effective*, which means that they are Poincaré dual to some holomorphic hypersurfaces.

**Question 1:** why is that true ?

We call a homology class in $H_2(M,\mathbb{Z})$ *effective* if it has non-negative intersection indices with fundamental cycles of all compact holomorphic hypersurfaces in $M$.

**Question 2:** What kind of intersection index are we talking about here?

Would this definition be equivalent to saying that a homology class $A \in H_2(M,\mathbb{Z})$ is effective if for any compact holomorphic hypersurface $B \in H_{n-2}(M, \mathbb{Z})$, the natural pairing $\langle B, A \rangle$ is non-negative ?

From now on, we suppose that the cohomology class $[\omega]$ is integral, that is, it is the image of an element in $H_2(M,\mathbb{Z})$.

**Question 3:** Is there any relation between the quotient $\underset{A \text{ effective}}{\max} \langle c, A \rangle / \langle [\omega], A \rangle$ and the cup-length of the toric manifold $M$ ?

**Question 4:** In the article, the cohomology class $[\omega]$ is assumed to be *primitive*. What does this mean ?

Toric manifolds can actually be constructed via a process called *symplectic reduction*: given the linear action of the torus $(S^1)^n$ on $\mathbb{C}^n$, and a subtorus $\mathbb{T}^k \subset (S^1)^n$, one can choose a regular value $p \in \text{Lie}(\mathbb{T}^k)^* \simeq \mathbb{R}^k$ of the moment map $P$ associated with the $\mathbb{T}^k$ action on $\mathbb{C}^n$, and define a toric manifold a the quotient $M_p := P^{-1}(p) / \mathbb{T}^k$. Given this construction, there is an isomorphism
$$\mathbb{Z}^k \simeq H_2(M, \mathbb{Z}),$$ which identifies the set of effective homology classes with the intersection of the lattice $\mathbb{Z}^k$ with the first orthant in $\mathbb{R}^n \simeq \text{Lie}((S^1)^n)$.

**Question 5:** what is this isomorphism, and why does this identification hold ?