Expression for $\nabla_{JX} Y$ on a Kähler manifold

Question

Let $(M, \omega, g, J)$ be a Kähler manifold with symplectic form $\omega$, Riemannian metric $g$ and complex structure $J$.

I'm looking for a formula that gives an expression for $\nabla_{J X} Y$, where $\nabla$ is the Levi-Civita connection and $X, Y$ are (real) vector fields on $M$. Actually, what I want to calculate is $\nabla_{JX} T$ where $T$ is some arbitrary tensor (but I guess this can be reduced to the case where $T$ is a vector field by the Leibniz identity).

Answer 1

Here's one alternative expression.

As $\nabla$ is torsion-free, we have $\nabla_{JX}Y - \nabla_Y(JX) - [JX, Y] = 0$, so

$$\nabla_{JX}Y = \nabla_Y(JX) + [JX, Y] = (\nabla_YJ)X + J(\nabla_YX) + [JX, Y] = J(\nabla_YX) + [JX, Y]$$

since $\nabla J = 0$ as $g$ is Kähler.