Ricci form of a Kähler manifold ($\rho$)

For a Kähler manifold, the Ricci form is the real $(1,1)$-formwhere $F_{T^{1,0}X}$ is the Chern curvature of the holomorphic tangent bundle. The trace is essential when the complex dimension is greater than one: the full curvature is endomorphism-valued, whereas $\rho$ is an ordinary differential form. The local trace of Chern curvature identity gives the determinant formula.^[ref]

Equivalently, $\rho (U,V)=\mathrm{Ric}(JU,V)$, and $[\rho /(2\pi )]=c_1(T^{1,0}X)=-c_1(K_X)$. The Ricci form therefore expresses Ricci curvature in the language of the complex structure and represents the tangent First Chern class.

In complex dimension one, $\rho =K\omega =\frac{1}{2}\mathrm{Scal}\omega$. A flat complex torus has $\rho =0$. On Complex projective space with ${\omega }_{FS}=i\partial \bar{\partial }\log (1+|z{|}^2)$, the metric determinant is proportional to $(1+|z{|}^2{)}^{-(n+1)}$, so $\rho =(n+1){\omega }_{FS}$. This is a concrete positive Kähler-Einstein metric.