TITLE: Does the Lie derivative commute with $\partial$?
QUESTION [11 upvotes]: It is well-known that on a smooth manifold $M$, the Lie derivative commutes with the exterior derivative, i.e.
$${\cal L}_Xd\alpha=d{\cal L}_X\alpha$$
for any vector field $X$ and differential form $\alpha$.
If $M$ is a complex manifold, is there a similar result for the partial derivative
$${\cal L}_X\partial\alpha=\partial{\cal L}_X\alpha?$$
(Edit: By "similar" I mean maybe it does not hold in this form but there is nevertheless an analogous statement?)

REPLY [2 votes]: It's true if the complex structure is invariant under the vector field X. In that case: $\mathcal{L}_X \partial a = \mathcal{L}_X (1 - \imath J) da = (1 - \imath J)\mathcal{L}_X da = \partial \mathcal{L}_X a$. (Note: This is written with $a$ being a scalar. For a $(p,q)$ form you just need to pick a suitable projector.)