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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):

[tex]\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})[/tex]

[tex]\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})[/tex]

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

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

[tex]\nabla_\frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j}[/tex]

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

[tex]\nabla_\frac{\partial}{\partial z_i} \frac{\partial}{\partial z_j}[/tex]

What I mean is that

[tex]\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}[/tex]

so does this mean then that the following is true?

[tex]\nabla_\frac{\partial}{\partial z_i} \frac{\partial}{\partial z_j} = (A_{i j}^k + iB_{i j}^k)\frac{\partial}{\partial z_k}[/tex]

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

[tex]\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}[/tex]

because

[tex]\frac{\partial f}{\partial x_i} \neq \frac{\partial f}{\partial z_i}[/tex]

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.