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  "body": "\n## Statement\n\nLet $X$ be a Fano manifold, that is a compact complex manifold whose anticanonical line bundle $-K_X$ is ample.\nThen $X$ admits a [[Kähler–Einstein metric|6ETEBA8827]] if and only if the pair $(X, -K_X)$ is K-polystable.\n\nHere K-polystable means the following.\nA test configuration of $(X, -K_X)$ is a $\\C^*$-equivariant degeneration of $X$ with a line bundle extending a power of $-K_X$.\nFor every normal test configuration, the Donaldson–Futaki invariant is nonnegative.\nMoreover, it vanishes only for normal test configurations that are products of $X$ with $\\C$ twisted by a $\\C^*$-action on $X$.\n\n## Remarks\n\nThis form of the conjecture, for Fano manifolds, was proved by Xiuxiong Chen, Simon Donaldson and Song Sun.\nThe direction \"Kähler–Einstein implies K-polystable\" is the easier one.\nThe converse uses the continuity method with metrics having cone singularities along a divisor, and [[Gromov–Hausdorff limits|GT56RTG9X5]] of these metrics, which are shown to be algebraic varieties.\n\nExample: the blow-up of $\\C P^2$ at one point is Fano but has no Kähler–Einstein metric.\nIts automorphism group is not reductive, which rules out a Kähler–Einstein metric by Matsushima's theorem, and a test configuration coming from a $\\C^*$-action has negative Donaldson–Futaki invariant.\nBy contrast $\\C P^n$ with the Fubini–Study metric is Kähler–Einstein and K-polystable.\n\nThe general conjecture asks for a constant scalar curvature Kähler metric in $c_1(L)$ on a polarized manifold $(X, L)$, under a suitable notion of K-stability.\nThat general form is open, and the right stability notion for it is still discussed.\n\n## Explanations\n\n### L1 human\n\nFor manifolds with positive first Chern class, a Kähler–Einstein metric may or may not exist.\nThe theorem says that existence is equivalent to an algebraic condition called K-polystability.\nThis condition looks at all ways the manifold can degenerate and checks a sign on each of them.\n\n### L2 human\n\nThe negative and zero cases of the Kähler–Einstein problem follow from the [[Calabi conjecture|WJKA5YBTJ7]] methods with no obstruction.\nIn the Fano case there are obstructions, such as the Futaki invariant and Matsushima's theorem, and K-stability collects them all.\nThe statement links a nonlinear partial differential equation to geometric invariant theory, in the spirit of the Hitchin–Kobayashi correspondence for vector bundles.\nAfter the proof, much work has gone into deciding K-stability of explicit Fano manifolds by algebraic means.\n",
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