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).
For Fano manifolds, existence is governed by algebro-geometric stability. The Yau–Tian–Donaldson conjecture, now resolved in the
Generalizations of the concept include Kähler-Ricci solitons, which satisfy Ric(g) − ∇∇f = λ g for a potential function