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The Fubini-Study form is the standard Kähler form on complex projective space. With the integral normalization, $[{\omega }_{FS}/(2\pi )]=c_1(\mathcal{O}(1))$.

The two-form is associated with the Fubini-Study metric.

A Kähler-Einstein metric is a Kähler metric $g$ whose Ricci tensor satisfiesfor a real constant $\lambda$. Equivalently, its Ricci form satisfies $\rho =\lambda \omega$, where $\omega$ is the Kähler form. It combines the compatibility of complex and Riemannian geometry with an Einstein curvature condition.

In holomorphic coordinates, take $\omega =i\sum g_{j\bar{k}}d{z}^j\wedge d{\bar{z}}^k$. The Ricci form is $\rho =-i\partial \bar{\partial }\log \det (g_{j\bar{k}})$. A constant positive Hermitian matrix therefore gives $\rho =0$: the flat metric on ${\mathbb{C}}^n$, and its quotient by a full lattice, are Ricci-flat examples. The Fubini-Study metric on Complex projective space gives a positive-$\lambda$ example, with the numerical value depending on the metric normalization.

The equation is a nonlinear condition on a local Kähler potential, through the logarithmic determinant of its complex Hessian. This is why complex Monge-Ampère equations enter existence questions. Not every Kähler metric satisfies the Einstein condition.^[ref]

Kähler geometry studies complex manifolds with a Hermitian metric $g$ whose associated real two-form $\omega (X,Y)=g(JX,Y)$ is closed. Here $J$ is the complex structure. Equivalently, the Levi-Civita connection preserves $J$. Thus the same metric simultaneously supports complex geometry, Riemannian geometry and symplectic geometry.^[ref]

On ${\mathbb{C}}^n$, the Euclidean metric gives $\omega ={\sum }_{j}d{x}_j\wedge d{y}_j$, which is closed and nondegenerate. Locally one can express a Kähler form as $\omega =i\partial \bar{\partial }\phi$ for a real Kähler potential $\phi$, using a stated normalization. On Complex projective space, $\phi =\log (1+{\sum }_j|z_j{|}^2)$ gives the Fubini-Study metric in an affine chart. The potentials need not glue as one global function; their forms glue. This is how locally elementary differentiation can define nontrivial global geometry.

Let a compact Lie group $G$ act on a Kähler manifold $(M,\omega ,J,g)$ by holomorphic maps and isometries, with an equivariant moment map $\mu$. If $c$ is a regular value fixed by the coadjoint action and $G$ acts freely on ${\mu }^{-1}(c)$, the Kähler quotient isIt is a Kähler manifold of complex dimension ${\dim }_{\mathbb{C}}M-{\dim }_{\mathbb{R}}G$. Compactness supplies properness of the action. If $\iota$ includes the level and $\pi$ is the quotient projection, the reduced Kähler form satisfies ${\pi }^{\ast }{\omega }_c={\iota }^{\ast }\omega$. The metric and complex structure descend through the $g$-orthogonal complement of the orbit tangent spaces inside the level tangent spaces. This horizontal space is $J$-invariant; the reduction theorem proves that the descended structure is integrable.^[ref]

For scalar circle group multiplication on ${\mathbb{C}}^{n+1}$, use the standard symplectic form and convention ${\iota }_{X_{\xi }}\omega =d{\mu }^{\xi }$. Then $\mu (z)=-|z{|}^2/2$. Reducing at $c=-R^2/2$ divides the radius-$R$ sphere by its Hopf fibration and gives Complex projective space with the corresponding scaled Fubini-Study metric, as computed in Fubini-Study form from circle reduction.^[ref]