Chern connection

The Chern connection is the unique connection on a Hermitian holomorphic bundle compatible with both its metric and holomorphic structure. Its curvature has type $(1,1)$.

The Ricci form is the real curvature form $i{F}_{\nabla }$ of the Chern connection on $T^{1,0}X$. In complex dimension one, $\rho =K\omega =\frac{1}{2}\mathrm{Scal}\omega$.