Recently i was studying line bundles on $\mathbb{C}P^1$. Here is my confusion:
for any holomorphic map $f:\mathbb{C}P^1 \to M$, where $(M,E,\nabla)$ is a $r$-rank holomorphic vector bundle with a linear connection $\nabla$.
Then $f^*E$ split into direct sums of $r$ line bundles $L_1\oplus\cdots \oplus L_r$ (Why??? any reference to this?)
If c_1(E) is positive, since $f^*c_1(E)=c_1(L_1)+\cdots+c_1(L_r)$, then $c_1(L_i)$ is positive for some $i$.
So what does it mean a first chern class (as a cohomology class of $H^2(M)$) is positive?
and why does a sum of first chern classes is positive indicates some of them must be positive?
If there is any reference to this all, please tell me.
Best Answer
We say $c_1(E)$ is positive if, when regarded as an element of $H^2_{\text{dR}}(X)$, it can be represented by a Kähler form, i.e. a closed positive real $(1, 1)$-form. We say $c_1(E)$ is negative if $-c_1(E)$ is positive. We could also have $c_1(E) = 0$ in $H^2_{\text{dR}}(X)$, but in general $c_1(E)$ is neither positive, negative, nor zero.
In the case of $\mathbb{CP}^1$, the situation simplifies. First we have an isomorphism $\Phi : H^2(\mathbb{CP}^1; \mathbb{Z}) \to \mathbb{Z}$ given by the orientation on $\mathbb{CP}^1$ induced by the complex structure. Then $c_1(E) \in H^2(\mathbb{CP}^1; \mathbb{Z})$ is positive in the above sense if and only if $\Phi(c_1(E)) > 0$; similarly, $c_1(E)$ is negative if and only if $\Phi(c_1(E)) < 0$. In particular, every holomorphic line bundle over $\mathbb{CP}^1$ is of the form $\mathcal{O}(a)$ for some $a \in \mathbb{Z}$ and $\Phi(c_1(\mathcal{O}(a))) = a$, so $\mathcal{O}(a)$ is positive if and only if $a > 0$. By the Grothendieck lemma, every holomorphic vector bundle $V \to \mathbb{CP}^1$ splits as a sum of line bundles $V \cong \mathcal{O}(a_1)\oplus\dots\oplus\mathcal{O}(a_r)$. Then
\begin{align*} \Phi(c_1(V)) &= \Phi(c_1(\mathcal{O}(a_1)\oplus\dots\oplus\mathcal{O}(a_r)))\\ &= \Phi(c_1(\mathcal{O}(a_1)) + \dots + c_1(\mathcal{O}(a_r)))\\ &= \Phi(c_1(\mathcal{O}(a_1))) + \dots + \Phi(c_1(\mathcal{O}(a_r)))\\ &= a_1 + \dots + a_r. \end{align*}
So $c_1(V)$ is positive if and only if $a_1 + \dots + a_r > 0$; note, this implies that $a_i > 0$ for some $i$, and hence $c_1(\mathcal{O}(a_i))$ is positive. Now apply the above to the bundle $V = f^*E$ (the line bundles $L_j$ you mention are the $\mathcal{O}(a_j)$).
As for a reference, I would suggest Complex Geometry by Huybrechts.