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  "body": "\n## Statement\n\nLet $X$ be a [[complex analytic space|HXSZA44J5J]] whose underlying topological space is [[compact|0TVJZYB6SH]], and let $\\mathcal{F}$ be a [[coherent analytic sheaf|DSJET02WPR]] on $X$.\nThen for every integer $q \\ge 0$ the [[sheaf cohomology|PQ6Q143ARJ]] group $H^q(X, \\mathcal{F})$ is a finite-dimensional complex vector space.\n\n## Remarks\n\nThe proof uses two finite coverings of $X$ by open [[Stein spaces|Z4AMJS5KA6]], one relatively compact in the other.\nBy [[Cartan's theorem B|R7VKRV8AGQ]] both coverings compute $H^q(X, \\mathcal{F})$ by Čech cochains, which form Fréchet spaces.\nThe restriction map from the larger covering to the smaller one is a compact operator that induces an isomorphism on cohomology.\nA theorem of Schwartz on compact perturbations of surjective maps between Fréchet spaces then gives finite dimension.\n\nExample: on the projective line $\\mathbb{P}^1$ with $\\mathcal{F} = \\mathcal{O}$, one has $H^0 = \\C$, the constants, and $H^1 = 0$.\n\nCompactness is necessary.\nOn $X = \\C$, the space $H^0(\\C, \\mathcal{O})$ of entire functions is infinite-dimensional, since it contains $1, z, z^2, \\dots$.\n\n## Explanations\n\n### L1 human\n\nOn a compact complex space, the spaces of global holomorphic sections of a coherent sheaf, and the higher cohomology groups that measure obstructions, are all finite-dimensional.\nSo each of them is described by a single number, its dimension.\nCompactness is what keeps these spaces small.\n\n### L2 human\n\nThe theorem is the source of the numerical invariants of compact complex spaces, such as the dimensions of spaces of sections of line bundles and the arithmetic genus.\nIt makes the Euler characteristic $\\sum_q (-1)^q \\dim H^q(X, \\mathcal{F})$ of a coherent sheaf well defined on a compact complex space, the quantity computed by the Riemann–Roch theorem.\n",
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