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  "body": "\n## Statement\n\nLet $X$ be a compact [[Kähler manifold|X60JG9AEQJ]] of complex dimension $n$, and let $0 \\le k \\le 2n$.\nLet $\\Omega^p_X$ be the sheaf of holomorphic $p$-forms on $X$.\nLet $H^{p,q}(X)$ be the space of classes in $H^k(X, \\C)$ represented by a closed form of type $(p, q)$.\nThen:\n\n1. The [[singular cohomology|6N7BC70CX0]] with complex coefficients decomposes as a direct sum\n$$H^k(X, \\C) = \\bigoplus_{p + q = k} H^{p,q}(X).$$\n2. Complex conjugation on $H^k(X, \\C) = H^k(X, \\R) \\otimes \\C$ maps $H^{p,q}(X)$ onto $H^{q,p}(X)$.\n3. There are natural isomorphisms $H^{p,q}(X) \\cong H^q(X, \\Omega^p_X)$ with [[sheaf cohomology|PQ6Q143ARJ]], which is Dolbeault cohomology.\n\n## Remarks\n\nThe idea of the proof: by the [[Hodge theorem|A127WFVYKW]], each class in [[de Rham cohomology|GX74C07T0G]] has a unique harmonic representative for the Laplacian $\\Delta_d$.\nThe Kähler identities give $\\Delta_d = 2\\Delta_{\\bar\\partial}$, so $\\Delta_d$ preserves the type $(p, q)$ of forms, and the harmonic part of a form splits by type.\nThe $\\Delta_{\\bar\\partial}$-harmonic $(p,q)$-forms represent Dolbeault cohomology, which gives part 3; part 2 holds because $\\Delta_d$ is a real operator.\n\nExample: for a compact Riemann surface of genus $g$, $H^1(X, \\C) = H^{1,0} \\oplus H^{0,1}$ with $H^{1,0}$ the $g$-dimensional space of holomorphic $1$-forms, so $b_1 = 2g$.\n\nConsequence: the Hodge numbers $h^{p,q} = \\dim H^{p,q}$ satisfy $h^{p,q} = h^{q,p}$, so every odd Betti number $b_{2m+1} = \\sum_{p+q = 2m+1} h^{p,q}$ is even.\nThe Kähler hypothesis is needed: the Hopf surface, diffeomorphic to $S^1 \\times S^3$, is a compact complex manifold with $b_1 = 1$, so no such decomposition with symmetry exists on it.\n\n## Explanations\n\n### L1 human\n\nThe cohomology of a compact Kähler manifold, such as a smooth projective complex variety, splits into pieces labelled by pairs $(p, q)$.\nThe piece $(p, q)$ comes from forms with $p$ holomorphic and $q$ antiholomorphic differentials.\nThe pieces $(p, q)$ and $(q, p)$ are mirror images under complex conjugation.\n\n### L2 human\n\nThe decomposition makes $H^k(X, \\Z)$ modulo torsion into a Hodge structure of weight $k$, which is the starting point of period maps and of the Torelli problem.\nCombined with the hard Lefschetz theorem and the Hodge–Riemann bilinear relations, it gives polarized Hodge structures.\nIt also links topology to holomorphic invariants: for example $h^{1,0}$, the number of independent holomorphic $1$-forms, equals half the first Betti number.\n",
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