Let $E$ be a holomorphic vector bundle over $M$ which is a complex manifold. Let $D$ be a connection on it. So $D$ maps sections of $E$ to sections of $T^*M \otimes E$. Let $D'$ and $D''$ be the $(1,0)$ and $(0,1)$ parts of $D$.
Are $D'$ and $D''$ in their own right connections?
I believe the answer to this is no. Because $D(a.e) = d(a) \otimes e + a D(e)$ which gives $D'(a.e) + D''(a.e) = d(a) \otimes e + a(D'(e) + D''(e))$
This doesn't give Leibniz rule for $D'$ and $D''$ individually. Am I right?
Now let $K$ be a Hermitian metric.
Then there exits operators $\delta '$ and $\delta ''$ of types $(1,0)$ and $(0,1)$ such that $D' + \delta ''$ and $D'' + \delta'$ are K-compatible connections.
How do I produce such operators?