Let $\mathbb{CP}^n$ the complex projective space and $U_0:= \{[1: z_1:...:z_n] \in \mathbb{CP}^n \ \vert \ z_i \in \mathbb{C} \} \subset \mathbb{CP}^n$ a standard open subset wrt first cordinate.
Let $O(1) \to \mathbb{CP}^n$ the dual of the tautological line bundle: $O(1)= O(-1)^*$.
I have to show in an exercise that for the restriction of the first Chern class of $O(1)$ to $U_0$ this identity holds:
$$ c_1(O(1)) \vert _{[1:z_1:...:z_n]} = -\frac{i}{2 \pi} \overline{\partial} \ \partial \ log \|(1,z_1,...,z_n)\|^2 _{\mathbb{C}^{n+1}}$$
Here $\overline{\partial}, \partial$ are the Dolbeault operators and $\|(1,z_1,...,z_n)\|^2 = 1+ z \bar{z}$ the standard hermitian norm of $(1,z_1,...,z_n) \in \mathbb{C}^{n+1}$.
There is also following hint given: It is recommended to find a local holomorphic section of the restricted tautological bundle $O(-1) \vert _{U_0} \to U_0$. The induced Riemann matric (the pullback metric) is also called Fubini Study metric.
I have no idea now to finish this proof. Can somebody help me?
What I have already done:
For Chern classes we have a nice identity: $c_1(O(1))= c_1(O(-1)^*)=(-1) \cdot c_1(O(-1))$ (I think it's useful here since the hint suggested to work with $O(-1)$.)
Keeping the hint in mind we choose a non vanishing section
$$s: \mathbb{C}^n \cong U_0 \to O(-1) \vert _{U_0}, [1:Z_1:...:Z_n] \mapsto ([1:Z_1:...:Z_n], k(z) \cdot (1,Z_1,...,Z_n))$$
with $k(z): U_0 \to \mathbb{C}^*$. I guess that maybe that one with $k(z)=1$ does the job...
Since over $U_0$ that's an diffeomorphism and the Fubini Study metric is compatible with standard hermitian metric $h: \mathbb{C}^n \times \mathbb{C}^n \to \mathbb{C}^n$ we obtain the pullback metric $g:= s^* h$ on $O(-1) \vert _{U_0}$.
Next, the first Chern class coinsides up to a constant with the curvature $\Omega^{O(-1)}$ (a $(1,1)$-form) induced by connection $(\nabla^{O(-1)})$:
$c_1(O(-1)) =\frac{\Omega^{O(-1)}}{2 \pi i} =\frac{(\nabla^{O(-1)})^2}{2 \pi i}$
Locally the connection $\nabla^{O(-1)}=$ has structure $d+ \theta$ where $d= \partial + \overline{\partial}$ is the differential and $\theta $ a $(1,0)$-form on $U_0$. That's a basic fact that locally the curvature has structure
$\Omega^{O(-1)}= (\nabla^{O(-1)})^2= d \theta + \theta \wedge \theta$.
and we need to show
$$d \theta + \theta \wedge \theta =\overline{\partial} \partial log \|(1,z_1,...,z_n)\|^2 $$
From here I haven't any idea how to continue the proof. Could somebody help me how to show this claim? How are $\theta$ and section $s$ related to each other?