Connection, Curvature and Chern class in Kahler geometry

The connection and curvature terms of the Chern connection on a Kahler manifold are found by usual differential geometry methods on real manifolds.  However, in the former case, all real variables are replaced by complex variables $z$ and $\bar{z}$, and the Chern connection helps us reduce the overall computation amount.  It is particularly helpful when we are finding Chern classes by $\det(I+\frac{i}{2\pi}\Omega)$.

In the following, we will see how this method is applied to find curvatures and Chern classes on basic Kahler manifolds.