 Kähler–Einstein 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]

A Kähler-Einstein metric is a special type of Riemannian metric that arises in differential geometry and algebraic geometry. It is associated with Kähler manifolds, which are a class of complex manifolds with a compatible symplectic structure. A Kähler manifold is a complex manifold \( (M, J) \) equipped with a Kähler metric \( g \), which is a Riemannian metric that is both Hermitian and symplectic.