The holomorphic line bundle associated to a divisor

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the construction on an arbitrary Riemann surface below. Let $X$ be a Riemann surface and let $D={\sum }_p{n}_p[p]$ be a divisor on $X$ (Divisors, principal divisors and canonical divisors on a Riemann surface). Local finiteness gives every point a holomorphic coordinate neighborhood whose domain contains at most one point of $\mathrm{supp}D$. The indexed family of all such coordinate neighborhoods covers $X$. Since $X$ is second countable, ${\mathrm{A}\mathrm{C}}_{\omega }$ selects a countable subcover, retaining a coordinate chart for each member (Assuming countable choice, every second countable space is Lindelöf). This countable subcover selection is the only use of ${\mathrm{A}\mathrm{C}}_{\omega }$; full AC is not used. If $X$ is compact, compactness supplies a finite subcover and the construction uses only finite choice.

For each $i$, define a local meromorphic equation $f_i$ for $D$ by $f_i=1$ if $U_i$ misses $\mathrm{supp}D$, and by $f_i=(z_i-z_i(p){)}^{n_p}$ if $U_i$ contains its unique support point $p$. Thus $(f_i)=D{\mathrm{\mid }}_{U_i}$. On each overlap, the ratio $g_{ij}:=\frac{f_i}{f_j}$ is holomorphic and nowhere zero, since its divisor is zero there. These functions obey $g_{ij}{g}_{jk}=g_{ik}$. Regard multiplication by $g_{ij}$ as its real $2\times 2$ matrix. Holomorphic functions are smooth (Holomorphic functions are real analytic and smooth in their two real coordinates), so these matrices form a smooth $GL(2,\mathbb{R})$ cocycle; the cocycle construction gives a smooth real rank-two bundle, and the transitions preserve fibrewise multiplication by $i$ and are holomorphic. This is a holomorphic line bundle $\mathcal{O}(D)$ (Smooth manifolds and their smooth charts, Smooth vector bundles, rank, fibres, and trivial bundles, Vector bundle charts and transition functions, Construction of a vector bundle from a smooth cocycle, Holomorphic line bundles and meromorphic sections on a Riemann surface).

Write $e_i$ for its local holomorphic frame, with transition convention $e_j=g_{ij}{e}_i$ (Local and global frames of a vector bundle, Local frames and local trivializations are equivalent data). The local sections $s_D{\mathrm{\mid }}_{U_i}:=f_i{e}_i$ agree on overlaps because $f_j{e}_j=f_j{g}_{ij}{e}_i=f_i{e}_i$. Hence they define a meromorphic section $s_D$ of $\mathcal{O}(D)$, and its local coefficient $f_i$ shows $(s_D)=D$. For a nonzero meromorphic function $h$, the section $h{s}_D$ is holomorphic exactly when each local coefficient $h{f}_i$ is holomorphic, equivalently when $(h)+D\ge 0$; the zero function gives the zero holomorphic section. Thus $H^0(X,\mathcal{O}(D)):=\mathrm{\Gamma }(X,\mathcal{O}(D))\cong L(D),h⟼h{s}_D,$ and the meromorphic sections correspond to all meromorphic functions $h$, including zero. More locally, the sheaf of holomorphic sections is ${\mathcal{O}}_X(D)(V):=\{h\in \mathcal{M}(V):{\mathrm{ord}}_p(h)\ge -D(p)\text{ for every }p\in V\}$ for open $V\subseteq X$, where $\mathcal{M}(V)$ means functions meromorphic on each connected component and a zero germ has order $+\mathrm{\infty }$. This includes sections that vanish identically on some components of $V$, whose principal divisor is undefined. The local coefficient map $h\mapsto h{f}_i$ identifies these bounds with holomorphic sections and commutes with restrictions. In particular, ${\dim }_{\mathbb{C}}\mathrm{\Gamma }(X,\mathcal{O}(D))=\mathrm{\ell }(D)$.

The transition functions immediately give $\mathcal{O}(D+D^{\mathrm{\prime }})\cong \mathcal{O}(D)\otimes \mathcal{O}(D^{\mathrm{\prime }})$, $\mathcal{O}(-D)\cong \mathcal{O}(D{)}^{\ast }$, and $\mathcal{O}(0)\cong X\times \mathbb{C}$ (Dual and Hom vector bundles). If $(g)=D^{\mathrm{\prime }}-D$, multiplication by $1\mathrm{/}g$ maps ${\mathcal{O}}_X(D)$ isomorphically to ${\mathcal{O}}_X(D^{\mathrm{\prime }})$, because ${\mathrm{ord}}_p(h\mathrm{/}g)+D^{\mathrm{\prime }}(p)={\mathrm{ord}}_p(h)+D(p)$, including zero germs; on global sections it is the corrected isomorphism $L(D)\to L(D^{\mathrm{\prime }})$. For a canonical divisor $K_0=(\omega )$, the map $h\mapsto h\omega$ identifies $\mathcal{O}(K_0)$ with the canonical bundle $K={\mathrm{\Lambda }}^{1,0}{T}^{\ast }X$: its local coefficients are holomorphic exactly when ${\mathrm{ord}}_p(h)+K_0(p)\ge 0$ for every $p$, including zero germs, and dividing a holomorphic differential by $\omega$ gives the inverse. Its holomorphic sections are therefore exactly the holomorphic differentials (Meromorphic differentials, orders and residues).

Changing the countable cover or the local equations does not change the isomorphism class: on a common refinement, if $f_i$ and $f_j^{\mathrm{\prime }}$ are the two local equations, the map $e_i\mapsto (f_j^{\mathrm{\prime }}\mathrm{/}{f}_i)e_j^{\mathrm{\prime }}$ is holomorphic and sends $f_i{e}_i$ to $f_j^{\mathrm{\prime }}{e}_j^{\mathrm{\prime }}$. These maps agree on overlaps because their ratios telescope, so they glue to the canonical identification. For compact $X$ the finite-cover construction is choice-free; the only choice principle used in the general construction is ${\mathrm{A}\mathrm{C}}_{\omega }$, to obtain a countable trivializing cover from the coordinate-neighborhood cover.