Divisors, Riemann--Roch, and Duality

Let $X$ be a Riemann surface (Riemann surfaces and holomorphic atlases). A divisor on $X$ is a function $D:X\to \mathbb{Z}$ whose support $\mathrm{supp}D:=\{p:D(p)\mathrm{\ne }0\}$ is locally finite: every point has a neighbourhood meeting the support in only finitely many points. Write $D={\sum }_{p\in X}D(p)[p]$. Divisors form the abelian group $\mathrm{Div}(X)$ under pointwise addition; $D$ is effective, written $D\ge 0$, if every coefficient is nonnegative, and $D\ge D^{\mathrm{\prime }}$ means $D-D^{\mathrm{\prime }}\ge 0$. If $X$ is compact, local finiteness and compactness imply that every divisor has finite support, and its degree is $\mathrm{deg}D:={\sum }_{p}D(p)\in \mathbb{Z}$.

For a meromorphic function $f$ near $p$, define ${\mathrm{ord}}_p(f)=+\mathrm{\infty }$ if its germ at $p$ is identically zero. Otherwise, in a coordinate $z$ with $z(p)=0$, write uniquely $f=z^{m}u$ with $m\in \mathbb{Z}$ and $u$ holomorphic and nonzero at $p$, and put ${\mathrm{ord}}_p(f):=m$. The value is independent of the coordinate, since a change of coordinate has the form $z^{\mathrm{\prime }}=zv(z)$ with $v(0)\mathrm{\ne }0$. For a nonzero meromorphic function on connected $X$, no germ is identically zero: the set of points where it vanishes on a neighbourhood is open and closed, by the local identity theorem after clearing any pole, so connectedness makes that set either empty or all of $X$ (Identity theorem for holomorphic functions). Its zeros and poles are isolated, so $(f):={\sum }_{p\in X}{\mathrm{ord}}_p(f)[p]$ is a divisor, called the principal divisor of $f$. At a zero of $f$ its order is the ramification index $e_p(f)$ of the map $f:X\to \widehat{\mathbb{C}}$ at $0$; at a pole it is $-e_p(f)$, using the target coordinate $1\mathrm{/}w$ at $\mathrm{\infty }$ (Holomorphic maps and meromorphic functions on Riemann surfaces, Local power-map normal form on Riemann surfaces, Ramification index, ramification order and branch value, Isolated singularities: removable, poles, and essential singularities). For nonzero meromorphic functions $f,g$, local orders add, so $(fg)=(f)+(g)$ and $(1\mathrm{/}f)=-(f)$.

When $X$ is compact, every nonconstant meromorphic function $f$ has degree-zero principal divisor. Indeed, $f:X\to \widehat{\mathbb{C}}$ is continuous and the target is Hausdorff because it is the Riemann sphere (Holomorphic maps and meromorphic functions on Riemann surfaces, Riemann surfaces and holomorphic atlases); if $K\subseteq \widehat{\mathbb{C}}$ is compact, then $K$ is closed, so $f^{-1}(K)$ is closed in compact $X$ and hence compact. Thus $f$ is proper. The degree theorem for proper holomorphic maps says that the sum of local degrees over each fibre is the same integer $d$ (In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact, Degree of a proper holomorphic map of Riemann surfaces). Applied to the fibres over $0$ and $\mathrm{\infty }$, this gives ${\sum }_{f(p)=0}{\mathrm{ord}}_p(f)=d={\sum }_{f(p)=\mathrm{\infty }}-{\mathrm{ord}}_p(f)$, hence $\mathrm{deg}((f))=0$; a nonzero constant also has divisor $0$. Two divisors are linearly equivalent, written $D\sim D^{\mathrm{\prime }}$, when $D-D^{\mathrm{\prime }}$ is principal.

For a nonzero meromorphic differential $\omega$ on $X$, the order ${\mathrm{ord}}_p(\omega )$ is the order of its local meromorphic coefficient in a coordinate. This is coordinate-independent because the transition factor for a differential is the derivative of a coordinate change, a holomorphic unit. The canonical divisor of $\omega$ is $(\omega ):={\sum }_p{\mathrm{ord}}_p(\omega )[p]$. If ${\omega }_1,{\omega }_2$ are nonzero meromorphic differentials, the local quotients of their coefficients glue to a nonzero meromorphic function $f={\omega }_1\mathrm{/}{\omega }_2$, and $({\omega }_1)=(f)+({\omega }_2)$; therefore all canonical divisors are linearly equivalent (Meromorphic differentials, orders and residues).

For a divisor $D$, set $L(D):=\{0\}\cup \{f\in \mathcal{M}(X):f\mathrm{\ne }0\text{ and }(f)+D\ge 0\},$ where $\mathcal{M}(X)$ is the field of meromorphic functions on $X$. Equivalently, a nonzero $f$ lies in $L(D)$ exactly when ${\mathrm{ord}}_p(f)\ge -D(p)$ for every $p$. These local lower bounds are preserved under addition and scalar multiplication, so $L(D)$ is a $\mathbb{C}$-vector space. Put $\mathrm{\ell }(D):={\dim }_{\mathbb{C}}L(D)\in {\mathbb{N}}_0\cup \{\mathrm{\infty }\}$. If $D\sim D^{\mathrm{\prime }}$ and $g$ is a nonzero meromorphic function with $(g)=D^{\mathrm{\prime }}-D$, then multiplication by $1\mathrm{/}g$ gives an isomorphism $L(D)\to L(D^{\mathrm{\prime }})$, because for $f\in L(D)$, $(f\mathrm{/}g)+D^{\mathrm{\prime }}=(f)-(g)+D^{\mathrm{\prime }}=(f)+D\ge 0;$ its inverse is multiplication by $g$. The meromorphic functions on $X$ form a field under the usual local sum, product and reciprocal operations (Meromorphic functions on a plane domain). In particular, when $X$ is compact and $\mathrm{deg}D<0$, the space $L(D)$ is zero: a nonzero $f\in L(D)$ would make the effective divisor $(f)+D$ have degree $\mathrm{deg}((f))+\mathrm{deg}D=\mathrm{deg}D<0$, impossible.

Assume the Axiom of Choice. Let $M$ be a smooth manifold (Smooth manifolds and their smooth charts), let $\mathcal{A}={\mathcal{C}}_M^{\mathrm{\infty }}$ be its sheaf of real-valued smooth functions, and let $\mathcal{F}$ be a sheaf of $\mathcal{A}$-modules. Then for every open $U\subseteq M$ and every $q>0$, $H^q(U,\mathcal{F}{\mathrm{\mid }}_U)=0.$ The Axiom of Choice supplies injective resolutions for module sheaves and for abelian-sheaf cohomology. Its consequences ${\mathrm{A}\mathrm{C}}_{\omega }$ and DC supply, respectively, the partitions of unity and the acyclic-resolution comparison used below; no stronger choice principle is used.

For an integer $n\ge 1$, complex projective space is the set of complex lines in ${\mathbb{C}}^{n+1}$ with the quotient topology ${\mathbb{P}}^n(\mathbb{C}):=({\mathbb{C}}^{n+1}\setminus \{0\})\mathrm{/}{\mathbb{C}}^{\times },[z_0:\cdots :z_n]:={\mathbb{C}}^{\times }(z_0,\dots ,z_n)\mathrm{.}$ The quotient projection is continuous and surjective. Its restriction to the unit sphere $S^{2n+1}\subset {\mathbb{C}}^{n+1}$ is still surjective, so ${\mathbb{P}}^n(\mathbb{C})$ is compact. It is Hausdorff: the map sending a nonzero vector $z$ to the orthogonal projection $z{z}^{\ast }\mathrm{/}\parallel z{\parallel }^2$ is continuous and constant on each complex line, hence descends by the quotient topology to a continuous injection from ${\mathbb{P}}^n(\mathbb{C})$ into the Hausdorff space of complex matrices. A continuous injection from a compact space to a Hausdorff space is a homeomorphism onto its image.

For open $V\subseteq {\mathbb{C}}^r$, a map $H:V\to {\mathbb{C}}^s$ is holomorphic if at every $a\in V$ there is a complex-linear map $A_a:{\mathbb{C}}^r\to {\mathbb{C}}^s$ such that $H(a+h)=H(a)+A_{a}h+R_a(h)$ with $\parallel R_a(h)\parallel \mathrm{/}\parallel h\parallel \to 0$ as $h\to 0$, $h\mathrm{\ne }0$. When $r=1$, this is equivalent to each component being holomorphic in the sense of Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions, since a finite vector of scalar difference quotients converges exactly when each component does. A complex atlas consists of charts into ${\mathbb{C}}^n$ with holomorphic transitions in both directions.

For $j\in \{0,\dots ,n\}$ let $U_j:=\{[z_0:\cdots :z_n]:z_j\mathrm{\ne }0\}$. These open sets cover projective space: their inverse images under the quotient projection are the open saturated sets where $z_j\mathrm{\ne }0$. The standard chart is ${\phi }_j:U_j⟶{\mathbb{C}}^n,[z_0:\cdots :z_n]⟼(z_0\mathrm{/}{z}_j,\dots ,\widehat{z_j\mathrm{/}{z}_j},\dots ,z_n\mathrm{/}{z}_j),$ where the $j$th coordinate is omitted. It is well defined under rescaling. The coordinate-ratio map on the inverse image of $U_j$ is continuous and constant on each quotient fibre, so the quotient topology makes ${\phi }_j$ continuous; its inverse inserts $1$ in the $j$th position and is continuous by composition with the quotient projection. Hence it is a homeomorphism. On $U_j\cap U_k$ the transition from the $j$th chart to the $k$th chart sends each coordinate $w_{\mathrm{\ell }}=z_{\mathrm{\ell }}\mathrm{/}{z}_j$ ($\mathrm{\ell }\mathrm{\ne }j$) to $w_{\mathrm{\ell }}\mathrm{/}{w}_k$ for $\mathrm{\ell }\mathrm{\ne }k$, with the omitted $j$th coordinate equal to $1\mathrm{/}{w}_k$. To check holomorphy at $a$ with $a_k\mathrm{\ne }0$, set $a_j=1$ and $h_j=0$ for the omitted source coordinate. Each target coordinate has the expansion $(a_{\mathrm{\ell }}+h_{\mathrm{\ell }})\mathrm{/}(a_k+h_k)=a_{\mathrm{\ell }}\mathrm{/}{a}_k+h_{\mathrm{\ell }}\mathrm{/}{a}_k-a_{\mathrm{\ell }}{h}_k\mathrm{/}{a}_k^2+O(\parallel h{\parallel }^2)$ for $\mathrm{\ell }\mathrm{\ne }k$. The linear term is complex-linear, and the remainder estimate follows by expanding the reciprocal at the nonzero $a_k$. The inverse transition has the same form, so both are holomorphic. Thus these charts make ${\mathbb{P}}^n(\mathbb{C})$ a complex manifold of complex dimension $n$ and a topological manifold of real dimension $2n$ in the sense of Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces. The finite chart cover by second-countable copies of ${\mathbb{C}}^n$ gives a countable base. Its rational transition functions have smooth real coordinate expressions on their domains, so the atlas also defines the smooth structure in the sense of Smooth manifolds and their smooth charts. The space is path-connected: distinct lines represented by $u,v$ give the path $t\mapsto [(1-t)u+tv]$, whose vector never vanishes because $u,v$ are linearly independent; equal lines give a constant path. In particular, ${\mathbb{P}}^1(\mathbb{C})$ is a Riemann surface in the sense of Riemann surfaces and holomorphic atlases.

A map $F:X\to {\mathbb{P}}^n(\mathbb{C})$ from a Riemann surface is holomorphic when it is continuous and every component of each chart expression ${\phi }_j\circ F$, written in a local coordinate of $X$, is holomorphic on the open set $F^{-1}(U_j)$. On an overlap, the new components are $f_{\mathrm{\ell }}\mathrm{/}{f}_k$ and $1\mathrm{/}{f}_k$, where the $f_{\mathrm{\ell }}$ are the old components and $f_k\mathrm{\ne }0$. The scalar quotient rule gives their holomorphy (Linearity, product, reciprocal, and quotient rules for complex derivatives), so it suffices to check one target chart locally. Changing the source coordinate composes each scalar component with a one-variable holomorphic chart transition, where The chain rule for complex derivatives applies (Holomorphic maps and meromorphic functions on Riemann surfaces). A projective line is the image in ${\mathbb{P}}^n(\mathbb{C})$ of a two-dimensional complex subspace of ${\mathbb{C}}^{n+1}$. Every invertible linear map of ${\mathbb{C}}^{n+1}$ induces a holomorphic projective linear transformation: in source and target standard charts its coordinates are ratios of affine-linear functions with nonzero denominator, and the same reciprocal expansion gives a complex-linear derivative; these transformations act transitively on ${\mathbb{P}}^n(\mathbb{C})$.

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.

Let $X$ be a Riemann surface and $E\to X$ a holomorphic line bundle (Riemann surfaces and holomorphic atlases, Holomorphic line bundles and meromorphic sections on a Riemann surface). For every open $U\subseteq X$, let ${\mathcal{O}}_X(E)(U)$ be the $\mathbb{C}$-vector space of holomorphic sections of $E{\mathrm{\mid }}_U$, with the usual restriction maps. Compatible local sections glue uniquely as sections of a bundle, so these groups form a sheaf of $\mathbb{C}$-vector spaces (A sheaf on a topological space, Presheaves and sheaves of groups, rings, and modules, Sections, restrictions, and global sections of a presheaf). Holomorphic local coefficients are smooth, so ${\mathcal{O}}_X(E)$ is a subsheaf of the sheaf of smooth sections (Subsheaves, Smooth sections, local sections, and support, Smoothness of a section is equivalent to smooth local components, Holomorphic functions are real analytic and smooth in their two real coordinates). When $E=\mathcal{O}(D)$, its global sections identify with $L(D)$ as in The holomorphic line bundle associated to a divisor; the general noncompact construction of $\mathcal{O}(D)$ uses ${\mathrm{A}\mathrm{C}}_{\omega }$, while its compact finite-cover construction is choice-free.

A finite good cover of $X$ is a finite indexed open cover $\mathfrak{U}=(U_0,\dots ,U_n)$ by holomorphic chart domains, each biholomorphic to a disc, such that every nonempty finite intersection of its members is also biholomorphic to a disc. This definition applies to a supplied finite good cover; it does not assert that every Riemann surface admits one.

For a supplied finite good cover $\mathfrak{U}$, define the ordered Čech cochain complex $C^{\bullet }(\mathfrak{U},{\mathcal{O}}_X(E))$ and its differential $\delta$ as in Ordered Čech cochain complex of a cover. Its degree-$p$ cocycles and coboundaries are $Z^p=\ker {\delta }^p$ and $B^p=\mathrm{im}{\delta }^{p-1}$, and the fixed-cover Čech cohomology is ${\check{H}}^p(\mathfrak{U},{\mathcal{O}}_X(E)):=Z^p\mathrm{/}{B}^p$ as in Fixed-cover Čech cohomology. In degree zero, restriction identifies ${\check{H}}^0(\mathfrak{U},{\mathcal{O}}_X(E))$ with $\mathrm{\Gamma }(X,E)$ (Čech H0 equals global sections).

If a finite good cover $\mathfrak{V}=(V_j)$ refines $\mathfrak{U}=(U_i)$ by a refinement function $c$ with $V_j\subseteq U_{c(j)}$, restriction defines a cochain map and hence a map on fixed-cover Čech cohomology (Refinement map of ordered open covers). The induced map on cohomology is independent of the chosen refinement function (Refinement choices induce the same Čech map). For a supplied pair of covers and refinement function, these Čech definitions use no Choice principle; the separate ${\mathrm{A}\mathrm{C}}_{\omega }$ input above pertains only to constructing the general noncompact divisor bundle.

Let $X$ be a compact Riemann surface, let $D$ be a divisor on $X$, put $E={\mathcal{O}}_X(D)$, and let $V\subseteq H^0(X,E)=L(D)$ be a complex vector subspace of dimension $N+1\ge 2$ (Divisors, principal divisors and canonical divisors on a Riemann surface, The holomorphic line bundle associated to a divisor). Assume $V$ is base-point-free: for every $p\in X$, some $s\in V$ has $s(p)\mathrm{\ne }0$ in the fiber $E_p$. Then evaluation ${\mathrm{ev}}_p:V\to E_p$ is surjective, and its dual embeds the one-dimensional space $E_p^{\ast }$ into $V^{\ast }$. The resulting line in $V^{\ast }$ defines a canonical map ${\phi }_V:X⟶\mathbb{P}(V^{\ast }),p⟼\mathbb{P}(\mathrm{im}({\mathrm{ev}}_p^{\ast }))\mathrm{.}$ For any ordered basis $B=(s_0,\dots ,s_N)$ of $V$, its dual basis identifies $\mathbb{P}(V^{\ast })$ with ${\mathbb{P}}^N(\mathbb{C})$, and in a local frame $e$ of $E$ with $s_j=f_{j}e$ the coordinate expression is ${\phi }_B(p)=[f_0(p):\cdots :f_N(p)]$. This expression is well defined and holomorphic on all of $X$. If another basis is $s_k^{\mathrm{\prime }}={\sum }_j{a}_{kj}{s}_j$, then ${\phi }_{B^{\mathrm{\prime }}}=\mathbb{P}(A)\circ {\phi }_B$ for $A=(a_{kj})\in G{L}_{N+1}(\mathbb{C})$. Thus the intrinsic map and its image in $\mathbb{P}(V^{\ast })$ depend only on $V$; the image in a fixed coordinate copy of ${\mathbb{P}}^N$ is carried by the induced projective linear transformation and need not be the same subset. The linear system $\mathrm{\mid }V\mathrm{\mid }:=\{(h)+D:0\mathrm{\ne }h\in V\subseteq L(D)\}$ also depends only on $V$.

Let ${\mathcal{O}}_{\mathbb{P}(V^{\ast })}(1)$ be the dual of the tautological line bundle (for this convention, points of projective space are lines). Then there is a canonical holomorphic line-bundle isomorphism $\mathrm{\Psi }:E\stackrel{\sim }{\to }{\phi }_V^{\ast }{\mathcal{O}}_{\mathbb{P}(V^{\ast })}(1)$ that sends each $s_j$ to the pullback of the corresponding homogeneous coordinate section $Z_j$. If $V=H^0(X,E)$, write ${\phi }_D$ for this map.

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a compact Riemann surface, let $E$ be a holomorphic line bundle, and supply compatible metrics $g,h$ as in the maximal Dolbeault-operator datum (The maximal Dolbeault operator and its Hilbert adjoint on a compact Riemann surface). Write ${\bar{\mathrm{\partial }}}_E$ for the smooth bundle Dolbeault operator defined by Holomorphic line bundles and meromorphic sections on a Riemann surface; the maximal operator $\bar{D}$ restricts to it on smooth sections. Set ${\mathrm{\Omega }}^{0,q}(X,E):=C^{\mathrm{\infty }}(X,{\mathrm{\Lambda }}^{0,q}{T}^{\ast }X\otimes E),q=0,1.$ Then the sheaf sequence $0⟶{\mathcal{O}}_X(E)⟶{\mathcal{E}}^0(E)\stackrel{{\bar{\mathrm{\partial }}}_E}{\to }{\mathcal{E}}^{0,1}(E)⟶0$ is exact, where ${\mathcal{E}}^q(E)$ is the sheaf of smooth $E$-valued $(0,q)$ forms. Moreover, $H^1(X,{\mathcal{O}}_X(E))\cong \frac{{\mathrm{\Omega }}^{0,1}(X,E)}{{\bar{\mathrm{\partial }}}_E{\mathrm{\Omega }}^{0,0}(X,E)},H^0(X,{\mathcal{O}}_X(E))=H^0(X,E),H^q(X,{\mathcal{O}}_X(E))=0(q\ge 2)\mathrm{.}$ The global identifications are natural in holomorphic bundle maps. For the fixed-cover comparison, additionally let $\mathfrak{U}$ be a supplied finite good cover of $X$ subordinate to holomorphic frame domains for $E$ (Cech cohomology of holomorphic sections of a line bundle on finite good covers). For every $p\ge 0$, its canonical Leray comparison map is an isomorphism ${\phi }_{\mathfrak{U}}^p:{\check{H}}^p(\mathfrak{U},{\mathcal{O}}_X(E))\stackrel{\sim }{\to }{H}^p(X,{\mathcal{O}}_X(E))\mathrm{.}$ These identifications are canonical and compatible with refinement; any two such frame-subordinate finite good covers identify canonically through $H^p(X,{\mathcal{O}}_X(E))$. We normalize the degree-one Dolbeault identification by Forster's convention: if a holomorphic Čech cocycle has a smooth splitting $c_{ij}=b_j-b_i$, its sheaf comparison class corresponds to $[{\bar{\mathrm{\partial }}}_E{b}_i]$. This is the negative of the identification obtained directly from the Čech–Dolbeault total differential $\delta +(-1{)}^p{\bar{\mathrm{\partial }}}_E$; the sign is fixed here for the residue pairing. The displayed quotient is a quotient of smooth forms; it does not assert that the Hilbert-space cokernel of the full maximal operator has already been identified with it.

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a compact Riemann surface and $D$ a divisor on $X$ (Divisors, principal divisors and canonical divisors on a Riemann surface). Put $E={\mathcal{O}}_X(D)$ and supply a Hermitian metric $h$ on $E$ and a compatible Riemannian metric $g$ on $X$ as in Hermitian metric and $L^2$ pairing on a compact Riemann surface. Write ${\bar{\mathrm{\partial }}}_E$ for the smooth Dolbeault operator on $E$, and set ${\mathrm{\Omega }}^{0,q}(X,E)=C^{\mathrm{\infty }}(X,{\mathrm{\Lambda }}^{0,q}{T}^{\ast }X\otimes E)$ for $q=0,1$.

1. The Dolbeault cohomology spaces $H^{0,0}(X,E)=\ker ({\bar{\mathrm{\partial }}}_E:{\mathrm{\Omega }}^{0,0}(X,E)\to {\mathrm{\Omega }}^{0,1}(X,E)),H^{0,1}(X,E)={\mathrm{\Omega }}^{0,1}(X,E)\mathrm{/}{\bar{\mathrm{\partial }}}_E{\mathrm{\Omega }}^{0,0}(X,E)$ are finite-dimensional. The first is $H^0(X,E)=\mathrm{\Gamma }(X,E)$ and equals the harmonic space $H^{0,0}(E)$. Every class in the second has a unique harmonic representative in $H^{0,1}(E)$.

2. The sheaf cohomology spaces $H^0(X,{\mathcal{O}}_X(D))$ and $H^1(X,{\mathcal{O}}_X(D))$ are finite-dimensional, and $H^0(X,{\mathcal{O}}_X(D))\cong L(D)$. For every supplied finite good cover subordinate to holomorphic frame domains of $E$, the fixed-cover group ${\check{H}}^1(\mathfrak{U},{\mathcal{O}}_X(E))$ is also finite-dimensional. Define $\mathrm{\ell }(D):=\dim H^0(X,{\mathcal{O}}_X(D))=\dim L(D),i(D):=\dim H^1(X,{\mathcal{O}}_X(D)),\chi ({\mathcal{O}}_X(D)):=\mathrm{\ell }(D)-i(D)\mathrm{.}$ Then $\mathrm{\ell }(D)$ and $i(D)$ are nonnegative integers and $\chi ({\mathcal{O}}_X(D))$ is an integer, which need not be nonnegative.

3. If $D^{\mathrm{\prime }}$ is linearly equivalent to $D$, then $\mathrm{\ell }(D^{\mathrm{\prime }})=\mathrm{\ell }(D)$, $i(D^{\mathrm{\prime }})=i(D)$, and $\chi ({\mathcal{O}}_X(D^{\mathrm{\prime }}))=\chi ({\mathcal{O}}_X(D))$.