= Ricci form of a Kähler manifold
{c}
{title2=$\rho$}
= Ricci form
{c}
{synonym}
For a <Kähler manifold>, the Ricci form is the real $(1,1)$-form
$$
\rho=i\operatorname{tr}F_{T^{1,0}X}
=-i\partial\bar\partial\log\det(g_{j\bar k}),
$$
where $F_{T^{1,0}X}$ 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.<Équations d’Einstein, interactions fondamentales de la physique, courbure et variétés de Calabi-Yau by Jean-Pierre Demailly>[Slide 16 of 23]{ref}
Equivalently, $\rho(U,V)=\operatorname{Ric}(JU,V)$, 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, $\rho=K\omega=\tfrac12\operatorname{Scal}\,\omega$. 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>.
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