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  "body": "\n## Summary\n\nA Hodge structure of weight $n$ is a lattice, or a rational vector space, with extra data on its complexification.\nThe complexification splits into pieces $H^{p,q}$ with $p + q = n$, and complex conjugation exchanges $H^{p,q}$ and $H^{q,p}$.\nIt abstracts the structure that the [[Hodge decomposition|HABCJ4Y0SR]] puts on the cohomology of a compact Kähler manifold.\nHodge structures form an abelian category with tensor products.\nThey are the linear algebra data behind period maps, variations of Hodge structure and the [[Hodge conjecture|XAEEDWV3PM]].\n\n## Explanations\n\n### L1 human\n\nA Hodge structure is a lattice $H_\\Z$ together with a way to split the complex vector space $H_\\Z \\otimes \\C$ into pieces, labelled by pairs of integers.\nThe pieces must be arranged symmetrically under complex conjugation.\nThe first example comes from an elliptic curve $E = \\C / (\\Z + \\tau \\Z)$, with $\\Im \\tau \u003e 0$.\nIts first cohomology $H^1(E, \\Z)$ is a rank $2$ lattice, and over $\\C$ it splits into the line of $dz$ and the line of $d\\bar z$, a Hodge structure of weight $1$.\nThe position of the line of $dz$ relative to the lattice recovers $\\tau$, and so the curve.\n",
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