I study Andrei Moroianu's Lectures on Kähler Geometry and have some troubles to understand the proof of Lemma 7.1. The claim is that the tensor $h$ that gives some called Fubini–Study metric on the complex projective space is positive definite:
All we have to do to show that $h$ is positive definite is to prove is that the restriction of $h$ to some local holomorphic chart $\phi: U_j \to \mathbb{C}^m$ defined by $\hat{h}(X,Y) := i \partial \overline{\partial} u(X, JY)$, for all $X,Y \in T\mathbb{C}^m$ is positive definite:
The auxilary function $u: \mathbb{C}^m \to \mathbb{R}$ is defined by $u(z)= \log(1+ \vert z \vert^2)$. The caluculation at point $p=(r,0,...,0) \in \mathbb{C}^m$ in the proof shows
$$\tag{1} \partial \overline{\partial} \log(1+ \vert z \vert^2)= \frac{1}{(1+r^2)^2}(dz_1 \wedge d \overline{z_1} + (1+r^2) \cdot \sum_{i=2}^m dz_i \wedge d \overline{z_i}) $$
Here the $dz_i$ are generators of the dual space of $T^{1,0}\mathbb{C}^m$ and $d\overline{z_i}$ are generators of the dual space of $T^{0,1}\mathbb{C}^m$ where these spaces are eigenspaces of $i$ and $-i$ with respect the endomorphism $J$ on the compexified tangent bundle
$$(T\mathbb{C}^m)^{\mathbb{C}}= T\mathbb{C}^m \otimes_{\mathbb{C}} \mathbb{C}= T^{1,0}\mathbb{C}^m \oplus T^{0,1}\mathbb{C}^m$$
See 1.4 The complexified tangent bundle page 9. Obsverve that since $\mathbb{C}^m$ is already complex the endomorphism $J$ on $(T\mathbb{C}^m)^{\mathbb{C}}$ is simply the multiplication by imaginary $i$.
Question: Why if we choose some $X,Y \in T\mathbb{C}^m$ then
$$\tag{2} \hat{h}(X,Y)= \frac{2}{(1+r^2)^2} Re(X_1\overline{Y_1} +(1+r^2) \cdot \sum_{i=2}^m X_i \wedge d \overline{Y_i})$$
That is we have to insert $X$ and $JY$ in
$$\tag{3} \hat{h}= i \partial \overline{\partial} \log(1+ \vert z \vert^2) = \frac{i}{(1+r^2)^2}(dz_1 \wedge d \overline{z_1} + (1+r^2) \cdot \sum_{i=2}^m dz_i \wedge d \overline{z_i})$$
which we have calculated in $(1)$. I not understand why the result of this evaluation is (2) as stated in last line of the Lemma's proof. Where the $2$ and the $Re$ come from? Why it's well defined? The summands $dz_i \wedge d \overline{z_i}$ acts on $T^{1,0}\mathbb{C}^m \times T^{0,1}\mathbb{C}^m$, but $X, JY$ live in $T\mathbb{C}^m$.
The claim is that the action of $i \cdot dz_j \wedge d \overline{z_j}$ on the pair $(X, JY)$ is $Re(X_j \overline{Y_j})$. Clearly it 'sees' only the $j$-th components but the rest of what happens there is obscure to me.
Thinking simpler assume $m=1$. Then $X, JY \in T\mathbb{C}$. Why $i \cdot dz \wedge d \overline{z}(X, JY)= 2 \cdot Re(X \overline{Y})$?
Moroianu seems to be using the convention that $dz_i\wedge d\bar{z_i} = dz_i\otimes d\bar{z_i} - d\bar{z_i}\otimes dz_i$. With this in mind, we have
\begin{align*} (dz_i\wedge d\bar{z_i})(X, Y) &= (dz_i\otimes d\bar{z_i})(X, Y) - (d\bar{z_i}\otimes dz_i)(X, Y)\\ &= dz_i(X)d\bar{z_i}(Y) - d\bar{z_i}(X)dz_i(Y)\\ &= X_i\overline{Y_i} - \overline{X_i}Y_i\\ &= X_i\overline{Y_i} - \overline{X\overline{Y_i}}\\ &= 2\operatorname{Re}(X_i\overline{Y_i}). \end{align*}