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  "body": "\n## Summary\n\nThe canonical bundle $\\omega_X$ of a smooth variety $X$ of dimension $n$ is the line bundle of differential forms of top degree $n$.\nIt is the $n$-th exterior power of the [[sheaf of differentials|6VC6HQTFSA]].\nA divisor $K_X$ with $\\mathcal{O}_X(K_X) \\cong \\omega_X$ is a canonical divisor.\nThe canonical bundle appears in [[Serre duality|N3QPQATSRN]], in the [[adjunction formula|RYB6F7JFGV]] and in the [[Kodaira dimension|ZW00Q0SAGR]].\nIts sign governs the broad classification of varieties: Fano varieties have anti-ample $\\omega_X$, Calabi–Yau varieties trivial $\\omega_X$, and varieties of general type big $\\omega_X$.\n\n## Explanations\n\n### L1 human\n\nOn a surface, a differential form of top degree looks locally like $g \\, dx \\wedge dy$, a way to measure small areas.\nThe canonical bundle is the line bundle whose sections are such top-degree forms.\nOn a curve these are the forms $g \\, dz$.\nOn the projective line, the form $dz$ has a double pole at infinity, since $w = 1/z$ gives $dz = -dw / w^2$.\nSo the canonical bundle of $\\mathbb{P}^1$ has degree $-2$, and for a curve of genus $g$ its degree is $2g - 2$.\n",
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