I'm confused by the following statement in complex differential geometry:
Let $E$ be a holomorphic vector bundle and let $E'$ be a holomorphic sub-bundle. Fix a unitary connection $\nabla$ on $E$ associated to some hermitian metric.
We can compare the curvature $F_{\nabla}$ on $E$ to the curvature of the connection $\nabla'$ on the sub-bundle obtained by orthogonal projection with respect to the metric.
The statement I am confused by is that $F_{\nabla'} \leq F_{\nabla}.$ I.e. "curvature decreases in holomorphic sub-bundles". This statement can be found in Griffiths and Harris, and in many other standard sources.
I am confused for the following reason:
Consider the inclusion of vector bundles on $\mathbb{C} P^1$:
$0 \to O(1) \to O(1) \oplus O(-1).$
Since curvature is proportional to the first Chern class, and $c_1(O(1))> c(O(1) \oplus O(-1))=0,$ this seems to contradict the above principle.
What am I misunderstanding?
Note: I thought initially that my confusion was due to some normalization, but since one can also consider $0 \to O(-1) \to O(1) \oplus O(-1),$ it seems like my confusion is due to something else.