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.
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A Kähler-Einstein metric is a Kähler metric whose Ricci tensor satisfiesfor a real constant . Equivalently, its Ricci form satisfies , where is the Kähler form. It combines the compatibility of complex and Riemannian geometry with an Einstein curvature condition.
In holomorphic coordinates, take . The Ricci form is . A constant positive Hermitian matrix therefore gives : the flat metric on , and its quotient by a full lattice, are Ricci-flat examples. The Fubini-Study metric on Complex projective space gives a positive- 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]
Lectures on Kähler Geometry by Andrei Moroianu. Sections 11 and 13, Ricci form and Kähler-Einstein condition