Relationship between Chern and Levi-Civita Connections on Kahler Manifolds

[cogito²]

So I'm trying to understand the statement: On a complex manifold with a hermitian metric the Levi-Civita connection on the real tangent space and the Chern connection on the holomorphic tangent space coincide iff the metric is Kahler.

I basically understand the meaning of this statement, but I'm becoming incredibly confused by the details. The isomorphism between the real tangent space and the holomorphic tangent space is basically just (as far as I've understood it):

$$\frac{\partial}{\partial x_i} \mapsto \frac{\partial}{\partial z_i} = \frac{1}{2}(\frac{\partial}{\partial x_i} - i\frac{\partial}{\partial y_i})$$

$$\frac{\partial}{\partial y_i} \mapsto i\frac{\partial}{\partial z_i} = \frac{1}{2}(\frac{\partial}{\partial x_i} + i\frac{\partial}{\partial y_i})$$

with the holomorphic space viewed as sitting inside the complexified tangent space.

So now what I wonder is the following. What exactly does

$$\nabla_\frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j}$$

correspond to in terms of the Chern connection? Is it simply this?

$$\nabla_\frac{\partial}{\partial z_i} \frac{\partial}{\partial z_j}$$

What I mean is that

$$\nabla_\frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j} = A_{i j}^k \frac{\partial}{\partial x_k} + B_{i j}^k \frac{\partial}{\partial y_k}$$

so does this mean then that the following is true?

$$\nabla_\frac{\partial}{\partial z_i} \frac{\partial}{\partial z_j} = (A_{i j}^k + iB_{i j}^k)\frac{\partial}{\partial z_k}$$

If that we're the case, then it would seem like

$$\nabla_\frac{\partial}{\partial x_i} f\frac{\partial}{\partial x_j} \neq \nabla_\frac{\partial}{\partial z_i} f\frac{\partial}{\partial z_j}$$

because

$$\frac{\partial f}{\partial x_i} \neq \frac{\partial f}{\partial z_i}$$

So I guess I'm just confused exactly what is meant in the books talking about this. I'm having trouble truly understanding the proofs I've found because I can't understand exactly what identifications are being made. If anyone can shed light on my problems or knows of a book that does this pretty explicitly I would be very thankful.

 

[cogito²]

I have thought about this some more and have become convinced that the following identification is correct:

Let $$X$$ be a complex manifold and $$(TX,J)$$ be the real tangent space with canonical complex structure. Then $$(TX,J)$$ is isomorphic to the holomorphic tangent bundle $$(T_X, i)$$ where the isomorphism connects the action of $$J$$ to that of $$i$$. Then say you have a connection $$\nabla$$ on $$TX$$ such that $$\nabla J = 0$$. Then you can put a real connection on $$T_X$$ which will preserve the action of $$i$$. I.e.:

$$\nabla : TX \times TX \to TX$$
$$\nabla_\frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j}$$

corresponds to:

$$\nabla : TX \times T_X \to T_X$$
$$\nabla_\frac{\partial}{\partial x_i} \frac{\partial}{\partial z_j}$$

Then because $$\nabla J = 0$$ you get

$$\nabla_\frac{\partial}{\partial x_i} i\frac{\partial}{\partial z_j} = i\nabla_\frac{\partial}{\partial x_i} \frac{\partial}{\partial z_j}$$

Then after that you can just extend

$$\nabla : TX \times T_X \to T_X$$

to

$$\nabla : TX \otimes_\mathbb{R} \mathbb{C} \times T_X \to T_X$$

linearly. Then this will be a complex connection. Now you can ask under which conditions the induced complex connection on $$T_X$$ will end up being a holomorphic connection. I've written up some fairly unilluminating conditions on the Christoffel symbols for this to be the case.

Now if you have a real metric $$g$$ on $$TX$$ and a hermitian metric $$h$$ on $$T_X$$ (and you'd probably like $$h$$ to be induced by $$g$$), you can ask whether under this process the Levi-Civita connection will induce the Chern connection.

I'm now fairly certain I am understanding this correctly, however if any of you see that I'm still doing something wrong, I'd appreciate the input! Also I'm still interested in seeing any source that does this clearly.