In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stability was first introduced by Gang Tian and reformulated more algebraically later by Simon Donaldson. The definition was inspired by a comparison to geometric invariant theory (GIT) stability. In the special case of Fano varieties, K-stability precisely characterises the existence of Kähler–Einstein metrics. More generally, on any compact complex manifold, K-stability is conjectured to be equivalent to the existence of constant scalar curvature Kähler metrics (cscK metrics).
History In 1954, Eugenio Calabi formulated a conjecture about the existence of Kähler metrics on compact Kähler manifolds, now known as the Calabi conjecture. One formulation of the conjecture is that a compact Kähler manifold X {\displaystyle X} admits a unique Kähler–Einstein metric in the class c 1 ( X ) {\displaystyle c_{1}(X)} . In the particular case where c 1 ( X ) = 0 {\displaystyle c_{1}(X)=0} , such a Kähler–Einstein metric would be Ricci flat, making the manifold a Calabi–Yau manifold. The Calabi conjecture was resolved in the case where c 1 ( X ) < 0 {\displaystyle c_{1}(X)<0} by Thierry Aubin and Shing-Tung Yau, and when c 1 ( X ) = 0 {\displaystyle c_{1}(X)=0} by Yau. In the case where c 1 ( X ) > 0 {\displaystyle c_{1}(X)>0} , that is when X {\displaystyle X} is a Fano manifold, a Kähler–Einstein metric does not always exist. Namely, it was known by work of Yozo Matsushima and André Lichnerowicz that a Kähler manifold with c 1 ( X ) > 0 {\displaystyle c_{1}(X)>0} can only admit a Kähler–Einstein metric if the Lie algebra H 0 ( X , T X ) {\displaystyle H^{0}(X,TX)} is reductive. However, it can be easily shown that the blow up of the complex projective plane at one point, Bl p C P 2 {\displaystyle {\text{Bl}}_{p}\mathbb {CP} ^{2}} is Fano, but does not have reductive Lie algebra. Thus not all Fano manifolds can admit Kähler–Einstein metrics. After the resolution of the Calabi conjecture for c 1 ( X ) ≤ 0 {\displaystyle c_{1}(X)\leq 0} attention turned to the loosely related problem of finding canonical metrics on vector bundles over complex manifolds. In 1983, Donaldson produced a new proof of the Narasimhan–Seshadri theorem. As proved by Donaldson, the theorem states that a holomorphic vector bundle over a compact Riemann surface is stable if and only if it corresponds to an irreducible unitary Yang–Mills connection. That is, a unitary connection which is a critical point of the Yang–Mills functional
YM ( ∇ ) = ∫ X ‖ F ∇ ‖ 2 d vol . {\displaystyle \operatorname {YM} (\nabla )=\int _{X}\|F_{\nabla }\|^{2}\,d\operatorname {vol} .}
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