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  "body": "\n## Definition\n\nLet $n$ be an integer.\nA *pure Hodge structure of weight $n$* consists of:\n\n1. a finitely generated [[abelian group|082GHNPJ53]] $H_\\Z$, with complex [[vector space|000AYPA9AJ]] $H_\\C = H_\\Z \\otimes_\\Z \\C$;\n2. a [[direct sum|0XYYT5KNTX]] decomposition $H_\\C = \\bigoplus_{p + q = n} H^{p,q}$ into complex subspaces;\n3. the symmetry $\\overline{H^{p,q}} = H^{q,p}$ for all $p, q$, where the bar is complex conjugation on $H_\\C$ with respect to the real form $H_\\Z \\otimes \\R$.\n\nA *rational Hodge structure* is defined in the same way with a finite-dimensional $\\Q$-vector space $H_\\Q$ in place of $H_\\Z$.\n\n## Remarks\n\nEquivalently, the data can be given by the Hodge [[filtration|HNK1Y62G7V]] $F^p = \\bigoplus_{p' \\ge p} H^{p', n - p'}$, a decreasing filtration of $H_\\C$ with $H_\\C = F^p \\oplus \\overline{F^{n-p+1}}$ for all $p$; then $H^{p,q} = F^p \\cap \\overline{F^q}$.\n\nExample: let $X$ be a compact Kähler manifold, for instance a smooth projective complex variety.\nThen $H^k(X, \\Z)$ modulo torsion, with the Hodge decomposition $H^k(X, \\C) = \\bigoplus H^{p,q}(X)$, is a Hodge structure of weight $k$: clauses 2 and 3 are parts 1 and 2 of the Hodge decomposition theorem.\nThe Tate structure $\\Z(1)$ is $H_\\Z = 2\\pi i \\Z \\subseteq \\C$ with $H_\\C = H^{-1,-1}$, a Hodge structure of weight $-2$.\n\nNon-example: let $H_\\Z = \\Z^2$ and $n = 1$, with $H^{1,0} = H_\\C$ and $H^{0,1} = 0$.\nClause 2 holds, but clause 3 fails, since $\\overline{H^{1,0}} = H_\\C \\ne 0 = H^{0,1}$.\n",
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