Kähler–Einstein metric

In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat.

The most important problem for this area is the existence of Kähler–Einstein metrics for compact Kähler manifolds. This problem can be split up into three cases dependent on the sign of the first Chern class of the Kähler manifold:

When first Chern class is not definite, or we have intermediate Kodaira dimension, then finding canonical metric remained as an open problem, which is called as Algebrization conjecture via Analytical Minimal Model Program.

A Kähler–Einstein manifold is one which combines the above properties of being Kähler and admitting an Einstein metric. The combination of these properties implies a simplification of the Einstein equation in terms of the complex structure. Namely, on a Kähler manifold one can define the Ricci form, a real , by the expression

Theorem (Yau): A compact Kähler manifold with trivial canonical bundle, a Calabi–Yau manifold, always admits a Kähler–Einstein metric, and in particular admits a Ricci-flat metric.

It was conjectured by Yau in 1993, in analogy with the similar problem of existence of Hermite–Einstein metrics on holomorphic vector bundles, that the obstruction to existence of a Kähler–Einstein metric should be equivalent to a certain algebro-geometric stability condition similar to slope stability of vector bundles.[11] In 1997 Tian Gang proposed a possible stability condition, which came to be known as K-stability.[12]

In this sense, the minimal model program can be viewed as an analogy of the Ricci flow in differential geometry, where regions where curvature concentrate are expanded or contracted in order to reduce the original Riemannian manifold to one with uniform curvature (precisely, to a new Riemannian manifold which has uniform Ricci curvature, which is to say an Einstein manifold). In the case of 3-manifolds, this was famously used by Grigori Perelman to prove the Poincaré conjecture.

A precise result along these lines was proven by Cascini and La Nave,[21] and around the same time by Tian–Zhang.[22]

It is possible to give an alternative proof of the Chen–Donaldson–Sun theorem on existence of Kähler–Einstein metrics on a smooth Fano manifold using the Kähler-Ricci flow, and this was carried out in 2018 by Chen–Sun–Wang.[24] Namely, if the Fano manifold is K-polystable, then the Kähler-Ricci flow exists for all time and converges to a Kähler–Einstein metric on the Fano manifold.

When the compact Kähler manifold satisfies the topological assumptions necessary for the Kähler–Einstein condition to make sense, the constant scalar curvature Kähler equation reduces to the Kähler–Einstein equation.

In the case where the holomorphic vector bundle is again the holomorphic tangent bundle and the Hermitian metric is the Kähler metric, the Hermite–Einstein equation reduces to the Kähler–Einstein equation. In general however, the geometry of the Kähler manifold is often fixed and only the bundle metric is allowed to vary, and this causes the Hermite–Einstein equation to be easier to study than the Kähler–Einstein equation in general. In particular, a complete algebro-geometric characterisation of the existence of solutions is given by the Kobayashi–Hitchin correspondence.