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  "body": "\n## Summary\n\nHodge theory studies the cohomology of smooth manifolds through harmonic differential forms.\nFor complex algebraic varieties, it studies the extra structure that harmonic forms put on cohomology.\nOn a compact Riemannian manifold, the [[Hodge theorem|A127WFVYKW]] identifies [[de Rham cohomology|GX74C07T0G]] with the space of harmonic forms.\nOn a compact [[Kähler manifold|X60JG9AEQJ]], harmonic forms split by type, which gives the [[Hodge decomposition|HABCJ4Y0SR]] and a [[Hodge structure|ZSM083M97C]] on each cohomology group.\nThe theory gives strong topological restrictions on algebraic varieties and leads to deep open problems such as the [[Hodge conjecture|XAEEDWV3PM]].\n\n## Explanations\n\n### L1 human\n\nA cohomology class on a manifold can be represented by many differential forms.\nHodge theory picks the best one, the harmonic form, which solves a Laplace equation and minimizes energy.\nOn complex manifolds with a compatible metric, harmonic forms can be sorted by how many $dz$ and $d\\bar{z}$ factors they contain.\nFor a compact Riemann surface of genus $g$, this sorts the $2g$-dimensional first cohomology into $g$ holomorphic forms and $g$ antiholomorphic ones.\nSuch facts constrain the shape of algebraic varieties, for instance the odd Betti numbers of a smooth projective variety are even.\n",
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