(Cartan--Serre)
Let \(X\) be a compact complex manifold, or more generally a compact complex space, and let \(\mathcal{F}\) be a coherent analytic sheaf on \(X\). Then \(H^i(X, \mathcal{F})\) is a finite-dimensional complex vector space for every \(i\).
(Serre's algebraic counterpart)
Let \(X\) be a projective scheme over a Noetherian ring \(A\) and \(\mathcal{F}\) a coherent sheaf on \(X\). Then each \(H^i(X, \mathcal{F})\) is a finitely generated \(A\)-module. With \(\OO_X(1)\) very ample, \(\mathcal{F}(m)\) is generated by global sections for \(m \gg 0\), and \(H^i(X, \mathcal{F}(m)) = 0\) for \(i > 0\) and \(m \gg 0\).
On a compact complex manifold the theorem gives finite-dimensionality of \(H^0(X, \Omega^p)\) and of the Dolbeault groups \(H^q(X, \Omega^p)\), and so the Hodge numbers \(h^{p,q}\) are defined. The analytic statements for Stein manifolds are Cartan’s Theorems A and B: every coherent analytic sheaf is generated by global sections at each stalk, and has vanishing cohomology in positive degrees.
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