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  "body": "\n## Problem\n\nLet $X$ be a connected [[complex manifold|JSGWAN2RJV]] and let $(U_i)_{i \\in I}$ be an open cover of $X$.\nFor each $i$, let $f_i$ be a [[meromorphic function|W6FTSF57T1]] on $U_i$ that is not identically zero on any component of $U_i$.\nAssume that $f_i / f_j$ is [[holomorphic|507C34W0ZT]] and nowhere zero on $U_i \\cap U_j$ for all $i, j$.\nFind a meromorphic function $f$ on $X$ such that $f / f_i$ is holomorphic and nowhere zero on $U_i$ for every $i$.\n\n## Remarks\n\nThe data prescribe the zeros and poles of $f$ with multiplicities, that is, a divisor on $X$.\nThe problem asks whether this divisor is the divisor of a global meromorphic function.\nThe quotients $g_{ij} = f_i / f_j$ form a Čech $1$-cocycle with values in the sheaf $\\mathcal{O}^*$ of nowhere-zero holomorphic functions, and they define a [[holomorphic line bundle|QAC7BYK9KX]].\nThe problem is solvable for these data if and only if this line bundle is holomorphically trivial.\nSo if $H^1(X, \\mathcal{O}^*) = 0$, every second Cousin problem on $X$ is solvable.\nOn a Stein manifold, Cartan's theorem B and the exponential sequence $0 \\to \\Z \\to \\mathcal{O} \\to \\mathcal{O}^* \\to 0$ give an isomorphism $H^1(X, \\mathcal{O}^*) \\cong H^2(X, \\Z)$.\nHence on a Stein manifold the problem is solvable if and only if the line bundle is topologically trivial, a result of Oka.\nIn particular it is always solvable on a Stein manifold with $H^2(X, \\Z) = 0$, for example on $\\C^n$ or on a convex domain.\nFor a domain in $\\C$ the problem is solved by the Weierstrass factorization theorem.\nThere are domains of holomorphy on which some second Cousin problem has no solution.\nAn example is $X = \\set{(z, w) \\in \\C^2 : (\\log \\abs{z})^2 + (\\log \\abs{w})^2 \u003c 1}$, a logarithmically convex Reinhardt domain that retracts onto the torus $\\abs{z} = \\abs{w} = 1$, so $H^2(X, \\Z) \\cong \\Z$.\nA holomorphic line bundle on $X$ with nonzero Chern class has a meromorphic section that is not identically zero, and its local expressions are second Cousin data without solution.\n\n## Explanations\n\n### L1 human\n\nIn one variable, Weierstrass showed how to build a holomorphic function with prescribed zeros and multiplicities.\nThe second Cousin problem asks the same in several variables, where zeros form hypersurfaces rather than isolated points.\nGiven local functions that define the wanted zeros and poles near each point, one wants one global function that defines them everywhere.\nUnlike the first Cousin problem, this one can fail even on very nice domains, for a topological reason.\nOka showed that on Stein manifolds topology is the only obstruction.\n",
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      "first_stated": "1895",
      "posed_by": "Pierre Cousin"
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    "msc": [
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    "name": "Second Cousin problem",
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        "url": "https://en.wikipedia.org/wiki/Cousin_problems"
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  "msc": [
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  "name": "Second Cousin problem",
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