Following this question, I would like to compute the coefficients of the connection form of the Chern connection of $\gamma^n$, the tautological line bundle over $\mathbb{C}P^n$, with respect to the natural Hermitian metric.
Given that $\gamma^n \subset \mathbb{C}P^n \times \mathbb{C}^{n+1}$, we can induce the metric from the standard Hermitian metric (dot product) on $\mathbb{C}^{n+1}$, where if $(p,v_i) \in \gamma^n$, then $h( (p,v_1), (p,v_2)) = v_1 \cdot v_2$.
Let $\nabla$ be the Chern connecion of $h$ over $\gamma^n$. How would I go about finding the connection form coefficients of $\nabla$?
From the linked question, I know that, locally, the connection form is $\frac{\bar{z} dz}{1+ |z|^2}$. But I do not know how to extract the coefficients from this, unless I'm missing the obvious.
EDIT: Perhaps to rephrase and clarify my question:
Given the Chern connection of the tautological bundle, what are its connection coefficients/Christoffel symbols?
Are they just $\frac{\bar{z}_k}{1+|z|^2}$ where $z = (z_1, \ldots, z_n)$ are local coordinates?
I believe I found a way to get the connection coefficients/Christoffel symbols. Please correct me if I am wrong.
Given the Chern connection $\nabla$ on $\gamma^n$, the bundle is a complex line bundle, so a rank $2$ real vector bundle, over a complex manifold of real dimension $2n$. I am using the real rank and dimension in order to count the Christoffel symbols properly, since they will be real functions.
So let $e_1, e_2$ be a (real) local frame for $\gamma^n$, $z_1, \ldots, z_n$ holomorphic local coordinates. I will split them as $z_k = x_k + i y_k$.
We have $\nabla e_1 = \omega_{1}^1 \otimes e_1 + \omega_{1}^2 \otimes e_2$ and $\nabla e_2 = \omega_{2}^1 \otimes e_1 + \omega_{2}^2 \otimes e_2$, where $\omega_{i}^{j}$ are the connection $1$-forms.
Now, we plug in the local basis vectors $\frac{\partial}{\partial x^k}, \frac{\partial}{\partial y^k}$ and get $\nabla_{\partial x^k} e_1 = \omega_{1}^1 (\frac{\partial}{\partial x^k})e_1 + \omega_{1}^2 (\frac{\partial}{\partial x^k})e_2$, etc. where $\omega_{i}^{j} (\frac{\partial}{\partial x^k}), \omega_{i}^{j} (\frac{\partial}{\partial y^k})$ are the connection coefficients/Christoffel symbols of $\nabla$. My count gives $8n$ coefficients.
Since in the holomorphic case we get the (complex) connection $1$-form to be $\frac{\bar{z}_k dz^k}{1+|z|^2}$ (Einstein summation), using the splitting $z_k = x_k + i y_k$ should naturally give the four real $1$-forms $$\omega_{1}^1 = \frac{x_k dx^k}{1+ |z|^2}, \omega_{1}^2 = \frac{x_k dy^k}{1+ |z|^2}, \omega_{2}^1 = -\frac{y_k dx^k}{1+ |z|^2}, \omega_{2}^2 = \frac{y_k dy^k}{1+ |z|^2}.$$
And thus, the $8n$ connection coefficients/Christoffel symbols should just be
\begin{align} \omega_{1}^1 (\frac{\partial}{\partial x^k}) &= \frac{x_k}{1+ |z|^2},\\ \omega_{1}^1 (\frac{\partial}{\partial y^k}) &= 0,\\ \omega_{1}^2 (\frac{\partial}{\partial x^k}) &= 0,\\ \omega_{1}^2 (\frac{\partial}{\partial y^k}) &= \frac{x_k}{1+ |z|^2},\\ \omega_{2}^1 (\frac{\partial}{\partial x^k}) &= -\frac{y_k}{1+ |z|^2},\\ \omega_{2}^1 (\frac{\partial}{\partial y^k}) &= 0,\\ \omega_{2}^2 (\frac{\partial}{\partial x^k}) &= 0,\\ \omega_{2}^2 (\frac{\partial}{\partial y^k}) &= \frac{y_k}{1+ |z|^2}. \end{align}
Does this work out?