The first chern class of Fano manifold

Question

If $M$ is a Fano manifold, $L$ is an ample line bundle over $M$. My question is that whether $c_1(L)=\alpha c_1(M)$ for some real number $\alpha$ always holds.

Answer 1

No, this is not true. The two classes $c_1(L)$ and $c_1(M)$ live in the abelian group $\operatorname{Pic}(M)$, but for many $M$ this group has rank larger than 1, and there are linearly independent ample classes in the group.

For an example, let $M$ be the blowup of $\mathbf P^2$ in a point. Then $\operatorname{Pic}(M)$ is isomorphic to $\mathbf Z^2$, generated by the hyperplane class $H$ and the exceptional divisor $E$. In this basis $c_1(M)=3H-E$, an ample class. But then for example if $L$ is a line bundle with class $c_1(L)=4H-E$, then $L$ is ample, but the Chern classes are linearly independent.

In fact one can show that ampleness is an open condition on line bundles, so whenever $\operatorname{Pic}(X)$ has rank bigger than 1, we can come up with a similar example.