Why the notation $\mathcal{O}(\mathcal{L})$ for line bundles $\mathcal{L}$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:29:09Z http://mathoverflow.net/feeds/question/86679 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86679/why-the-notation-mathcalo-mathcall-for-line-bundles-mathcall Why the notation $\mathcal{O}(\mathcal{L})$ for line bundles $\mathcal{L}$ AGBeginner 2012-01-26T01:07:55Z 2012-01-26T14:02:15Z <p>Let $X$ be a complex manifold. If $D$ is a divisor on $X$, then $\mathcal{O}(D)$ denotes the (up to isomorphic) line bundle $\mathcal{L}$ with a meromorphic section $m$ whose divisor $(m)=D$. </p> <p>But if $\mathcal{L}$ is a line bundle on $X$, there is also the notation $\mathcal{O}(\mathcal{L})$. Can anyone tell me what does it mean? Thank you very much. </p> http://mathoverflow.net/questions/86679/why-the-notation-mathcalo-mathcall-for-line-bundles-mathcall/86703#86703 Answer by diverietti for Why the notation $\mathcal{O}(\mathcal{L})$ for line bundles $\mathcal{L}$ diverietti 2012-01-26T09:13:39Z 2012-01-26T09:23:13Z <p>Slightly expanding the comment of J. C. Otterm, I think $\mathcal O_X(D)$ should not be interpreted as a line bundle but rather as a sheaf of sections.</p> <p>So, summing up:</p> <ul> <li><p>When you consider a (holomorphic) line bundle $L\to X$ you should think at that as a complex manifold together with a holomorphic surjective map to $X$, locally trivial, whose fibers are complex vector spaces of dimension one (and the local trivialization are compatible with the vector space structure).</p></li> <li><p>When you consider a (Weil or Cartier: I am assuming $X$ to be smooth so that the two concepts coincide) divisor $D$, you should look at it just as a formal integral combination of codimension one irreducible subvarieties.</p></li> <li><p>When you consider $\mathcal O_X(L)$, you should look at it as the sheaf of holomorphic sections of $L\to X$.</p></li> <li><p>When you consider $\mathcal O_X(D)$, if $D=\sum_j a_j D_j$ where $a_j\in\mathbb Z$ and $D_i$ are prime divisors, you should look at it as the sheaf of meromorphic function on $X$ which have at least zeros of order $a_i$ along $D_i$ if $a_i\le 0$ and at most poles of order $a_k$ along $D_k$ if $a_k\ge 0$.</p></li> </ul> <p>Of course, these four concepts are strongly related. </p> <p>For instance, given a divisor $D$ one can form an associated holomorphic line bundle, let's say $L_D$ and then consider its sheaf of holomorphic sections $\mathcal O_X(L_D)$ which is naturally isomorphic to $\mathcal O_X(D)$.</p> <p>On the other hand, given a holomorphic line bundle $L\to X$ where $X$ is projective, then it always admits a meromorphic section $\sigma$. Let $D_\sigma$ its associated divisor. Then, $L_{D_\sigma}\simeq L$ as holomorphic line bundles. </p>