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  "body": "\n## Statement\n\nLet $X$ be a smooth projective [[surface of general type|SV4CK2XP8M]] over $\\C$.\nLet $c_1^2 = K_X^2$ be the self-intersection of its [[canonical divisor|W05VYY4HBR]], and let $c_2$ be its second [[Chern number|PAD37GKDPD]], equal to the topological Euler characteristic of $X$.\nThen\n$$c_1^2 \\le 3 c_2.$$\n\n## Remarks\n\nMiyaoka and Yau proved the inequality independently, Miyaoka by algebraic methods and Yau by constructing Kähler–Einstein metrics; Bogomolov had earlier proved the weaker bound $c_1^2 \\le 4 c_2$.\nYau's proof shows that equality holds exactly when $X$ is a quotient of the unit ball in $\\C^2$ by a discrete group acting freely.\nExamples are the fake projective planes, with $c_1^2 = 9$ and $c_2 = 3$.\nExample: a smooth quintic surface in $\\mathbb{P}^3$ has $c_1^2 = 5$ and $c_2 = 55$, and $5 \\le 165$.\nThe hypothesis on the Kodaira dimension is needed: for $X = C \\times \\mathbb{P}^1$ with $C$ of genus $2$, one has $c_1^2 = 8(1 - 2) = -8$ and $c_2 = 4(1 - 2) = -4$, so $c_1^2 \u003e 3 c_2 = -12$.\nThe inequality fails in positive characteristic in general.\n\n## Explanations\n\n### L1 human\n\nTwo numbers measure a complex surface: the self-intersection of its canonical class and its Euler characteristic.\nFor surfaces of general type, the first is never more than three times the second.\nSo surfaces of general type cannot be too curved compared with their topology.\n\n### L2 human\n\nTogether with Noether's inequality $c_1^2 \\ge 2 \\chi(\\mathcal{O}_X) - 6$ for minimal surfaces, the inequality confines the Chern numbers of surfaces of general type to a wedge, the domain of the geography of surfaces.\nIt is the surface case of the Chern number inequalities for varieties with Kähler–Einstein metrics.\nYau used the equality case to prove that every compact complex surface homotopy equivalent to $\\mathbb{P}^2$ is isomorphic to $\\mathbb{P}^2$.\n",
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