Let $\mathbb{P}^n$ the projective space (or more general a Kähler manifold $X$) and
$$\kappa = \bigwedge^n \Omega_X$$
the canonical line bundle, the top exterior power of the bundle of holomorphic Kähler differentials.
The Levi-Civita connection corresponding to metric on $X$ gives rise to a connection on $\kappa $. The curvature of this connection is the two form defined by
$$ \rho(X ,Y) := Ric (JX,Y)$$
where $J$ is the complex structure map on the tangent bundle determined by the structure of the Kähler manifold and $Ric(-,-)$ the Ricci curvature . Claim: The Ricci form is a closed 2-form, whose cohomology class is, up to a real constant factor, the first Chern class of the canonical bundle.
Could somebody sketch the proof or provide lecture notes with a proof of this claim?