KählerEinstein

KählerEinstein refers to Kähler metrics on complex manifolds that satisfy the Einstein condition, meaning the Ricci curvature is proportional to the metric: Ric(g) = λ g for some real number λ. Equivalently, the Ricci form represents a multiple of the Kähler form, and the constant λ is linked to the first Chern class of the manifold. Such metrics are central objects in complex differential geometry and connect complex structure with curvature.

The existence and nature of KählerEinstein metrics depend on the sign of the first Chern class c1(M). If c1(M) < 0, Aubin and Yau showed that there exists a unique KählerEinstein metric in each Kähler class, obtained by solving a complex Monge–Ampère equation. If c1(M) = 0, Yau’s solution to the Calabi conjecture implies the existence of a unique Ricci-flat Kähler metric in every Kähler class, known as a Calabi–Yau metric. When c1(M) > 0, the manifold is Fano and the situation is more delicate; a KählerEinstein metric does not always exist.

For Fano manifolds, existence is governed by algebro-geometric stability. The Yau–Tian–Donaldson conjecture, now resolved in the last decade, states that a Fano manifold admits a KählerEinstein metric if and only if it is K-stable (with related conditions involving the Mabuchi K-energy). Obstructions such as the Futaki invariant can prevent existence. Many examples, including complex projective space and many homogeneous spaces, do admit KählerEinstein metrics, while others do not.

Generalizations of the concept include Kähler-Ricci solitons, which satisfy Ric(g) − ∇∇f = λ g for a potential function f and arise in the presence of holomorphic vector fields.