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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.

So, summing up:

• 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).

• 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.

• When you consider $\mathcal O_X(L)$, you should look at it as the sheaf of holomorphic sections of $L\to X$.

• 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 $a_k\ge 0$.

Of course, these four concepts are strongly related.

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 $\mathcal O_X(D)$.

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, $L_{D_\sigma}\simeq D_\sigma$ L$as holomorphic line bundles. 1 Slightly expanding the comment of J. C. Otterm, I think$\mathcal O_X(D)$should not be interpreted as a line bundle but rather as a sheaf of sections. So, summing up: • When you consider a (holomorphic) line bundle$L\to X$you should think at that as a complex manifold together with a holomorphic surjective map to$X$, locally trivial, whose fibers are complex vector spaces of dimension one (and the local trivialization are compatible with the vector space structure). • When you consider a (Weil or Cartier: I am assuming$X$to be smooth so that the two concepts coincide) divisor$D$, you should look at it just as a formal integral combination of codimension one irreducible subvarieties. • When you consider$\mathcal O_X(L)$, you should look at it as the sheaf of holomorphic sections of$L\to X$. • When you consider$\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$. Of course, these four concepts are strongly related. For instance, given a divisor$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)$. On the other hand, given a holomorphic line bundle$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 D_\sigma\$ as holomorphic line bundles.