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  "body": "\n## Definition\n\nLet $(M, J)$ be a complex manifold and let $g$ be a Kähler metric on $M$.\nThus $g$ is a [[Hermitian metric|CK1DCPC3D3]] whose fundamental form $\\omega$ is closed.\nLet $\\operatorname{Ric}$ be the [[Ricci curvature|4GTNM0WHDH]] of $g$, and let $\\rho(X, Y) = \\operatorname{Ric}(JX, Y)$ be its Ricci form.\nThe metric $g$ is *Kähler–Einstein* if there is a real constant $\\lambda$ such that $\\operatorname{Ric} = \\lambda g$, or equivalently $\\rho = \\lambda \\omega$.\n\n## Remarks\n\nExample: the flat metric on a complex torus $\\C^n / \\Lambda$, for a lattice $\\Lambda$ of rank $2n$.\nIt is Kähler with constant coefficients, its curvature vanishes, and so $\\operatorname{Ric} = 0 = 0 \\cdot g$.\nThe Fubini–Study metric on $\\C P^n$ is Kähler–Einstein with $\\lambda \u003e 0$, and the Bergman metric on the unit ball of $\\C^n$ is Kähler–Einstein with $\\lambda \u003c 0$.\n\nNon-example: on $\\C P^1 \\times \\Sigma$, with $\\Sigma$ a compact Riemann surface of genus $2$, take the product of a round metric of curvature $1$ and a metric of curvature $-1$.\nIt is Kähler, but $\\operatorname{Ric}$ equals $g$ on the first factor and $-g$ on the second, so no single $\\lambda$ works.\n\nOn a compact Kähler manifold the Ricci form represents $2\\pi c_1(M)$ in de Rham cohomology.\nSo a Kähler–Einstein metric with constant $\\lambda$ forces $2\\pi c_1(M) = \\lambda [\\omega]$: the first Chern class is negative, zero or positive according to the sign of $\\lambda$.\nA Kähler–Einstein metric is an [[Einstein metric|41S2EZ0KR3]] in the Riemannian sense.\n",
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