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  "body": "\n## Definition\n\nLet $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over an algebraically closed field, with structure sheaf $\\mathcal{O}_X$ and [[canonical bundle|W05VYY4HBR]] $\\omega_X$.\n$X$ is a *K3 surface* if:\n\n1. $\\omega_X \\cong \\mathcal{O}_X$, and\n2. the [[sheaf cohomology|PQ6Q143ARJ]] group $H^1(X, \\mathcal{O}_X)$ is $0$.\n\n## Remarks\n\nOver $\\C$, the same two conditions define K3 surfaces among all compact complex surfaces; such a surface need not be projective.\nA K3 surface has Hodge numbers $h^{2,0} = 1$ and $h^{1,1} = 20$, second Betti number $22$ and topological Euler characteristic $24$.\nExample: a smooth quartic $X \\subset \\mathbb{P}^3$ has $\\omega_X \\cong \\mathcal{O}_X(4 - 4) = \\mathcal{O}_X$ by adjunction, and $H^1(X, \\mathcal{O}_X) = 0$ from the exact sequence $0 \\to \\mathcal{O}_{\\mathbb{P}^3}(-4) \\to \\mathcal{O}_{\\mathbb{P}^3} \\to \\mathcal{O}_X \\to 0$ and the vanishing of $H^1$ and $H^2$ of line bundles on $\\mathbb{P}^3$.\nNon-example: an abelian surface, such as $E \\times E$ for an elliptic curve $E$, has trivial canonical bundle but $\\dim H^1(X, \\mathcal{O}_X) = 2$, so it fails clause 2.\nOther examples are double covers of $\\mathbb{P}^2$ branched along a smooth sextic and the Kummer surfaces, obtained from abelian surfaces by dividing by $x \\mapsto -x$ and resolving the $16$ singular points.\n",
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