Let $M$ be a Kähler manifold. We say that $M$ is Fano if the anticanonical line bundle $K_M^*$ of $M$ is ample (or positive).
On the other hand, I sometimes see the following definition (or sufficient condition) especially in the context of mirror symmetry.
Define the effective cone of $M$ by $$K(M) := \lbrace \phi_*[C] \in H_2(M,\mathbb Z) \mid C\ \text{curve}, \phi:C \to M \rbrace.$$ If the Kähler form $\omega$ is positive on $K(M)$, then $M$ is Fano.
Question: How do we see that the positivity of $\omega$ on $K(M)$ implies (or is equivalent to) the Fano property?
In particular I want to know the relation between the anticanonical bundle $K_M^*$ and the Kähler form $\omega$.