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  "body": "\n## Summary\n\nA Kähler–Einstein metric is a Kähler metric whose Ricci curvature is a constant multiple of the metric.\nIt is both a Kähler metric and an [[Einstein metric|41S2EZ0KR3]], so it is a canonical choice of metric on a complex manifold.\nOn a compact manifold its existence forces the first Chern class to be negative, zero or positive, and the three cases behave very differently.\nThe negative case was settled by Aubin and by Yau, and the zero case by Yau; the positive case leads to the [[Yau–Tian–Donaldson conjecture|DHYSPSYSCB]].\n\n## Explanations\n\n### L1 human\n\nOn a complex manifold there are many ways to measure lengths that respect the complex structure.\nA Kähler–Einstein metric is one whose curvature, in the Ricci sense, is the same in every direction at every point.\nThe round sphere, seen as the complex projective line, is an example with positive curvature.\nA flat torus is an example with zero curvature, and the hyperbolic disc is one with negative curvature.\n",
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