Are there compact analogues of Cartan's theorems A and B? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:35:35Z http://mathoverflow.net/feeds/question/37698 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37698/are-there-compact-analogues-of-cartans-theorems-a-and-b Are there compact analogues of Cartan's theorems A and B? Colin Tan 2010-09-04T06:15:10Z 2010-09-04T15:41:57Z <p>Cartan's theorem A says that on for a coherent sheaf ${\mathcal{F}}$ on a Stein manifold X, the fibres ${\mathcal{F}}_x$ over each point x in X are generated by global sections.</p> <p>I'm wondering if there are compact analogues of these theorem. Here I consider holomorphic line bundles over a compact complex manifold X. Consider the <strike> fibre</strike> stalk ${\mathcal{O}}_{X,x}$ of holomorphic germs over some point x. Is this <strike> fibre</strike> stalk generated by quotients of global holomorphic line bundle sections? That is, given two global sections $s$ and $t$ of the same line bundle $L\to X$. The quotient $s/t$ is a global meromorphic function. Given a holomoprhic function germ $f_x\in {\mathcal{O}}_{X,x}$, we can always find such $s$ and $t$ such that $(s/t)_x=f_x$? </p> <p>An analogue for Cartan's theorem B would be nice too. But I can't phrase this precisely.</p> http://mathoverflow.net/questions/37698/are-there-compact-analogues-of-cartans-theorems-a-and-b/37700#37700 Answer by Georges Elencwajg for Are there compact analogues of Cartan's theorems A and B? Georges Elencwajg 2010-09-04T07:35:01Z 2010-09-04T07:42:16Z <p>Dear Colin , for $X$ a holomorphic connected manifold, denote by $\mathcal M (X)$ its field of meromorphic functions.</p> <p>A) It is not true that a germ of holomorphic function $f_x\in \mathcal O_{X,x}$ is induced by a global meromorphic function : many compact complex manifolds only have $\mathbb C$ as meromorphic functions: $\mathcal M (X)=\mathbb C$. There is an example with $X$ a surface in Shafarevich's Basic Algebraic Geometry, volume 2, page 164.</p> <p>B) The best analogon to Theorem B is probably Cartan-Serre's result that for any coherent sheaf $\mathcal F$ on the compact manifold $X$, the cohomology vector spaces $H^q(X,\mathcal F), q\geq 1$ are finite-dimensional over $\mathbb C$.</p> <p>(Original article: Cartan-Serre, C.R.Acad.Sci. Paris <strong>237</strong> (1953), 128-130)</p> http://mathoverflow.net/questions/37698/are-there-compact-analogues-of-cartans-theorems-a-and-b/37731#37731 Answer by inkspot for Are there compact analogues of Cartan's theorems A and B? inkspot 2010-09-04T15:41:57Z 2010-09-04T15:41:57Z <p>Colin, I think you have answered your own question in your response to Brian Conrad. The fraction field of $\mathcal O_{X,x}$ has infinite transcendence degree over $\mathbb C$, while $\mathcal M(X)$, in Elencwajg's notation, has finite transcendence degree. For any global sections $s,t$ of a holomorphic line bundle, the ratio $s/t$ lies in $\mathcal M(X)$. So most $f$ cannot be written as $s/t$.</p>