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Concept
K-stability
Summary
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).
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 admits a unique Kähler–Einstein metric in the class . In the particular case where , 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 by Thierry Aubin and Shing-Tung Yau, and when by Yau. In the case where , that is when 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 can only admit a Kähler–Einstein metric if the Lie algebra is reductive. However, it can be easily shown that the blow up of the complex projective plane at one point, 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 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.
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