Cech--Dolbeault comparison for holomorphic line bundles on a compact Riemann surface

Let ${\mathcal{E}}^0(E)$ and ${\mathcal{E}}^{0,1}(E)$ denote the sheaves of smooth sections and smooth $E$-valued $(0,1)$-forms. In a holomorphic frame, ${\bar{\mathrm{\partial }}}_E(fe)=(\bar{\mathrm{\partial }}f)e$, so the kernel sheaf is ${\mathcal{O}}_X(E)$ by [F1]. For any point $x$, choose a holomorphic coordinate disc and frame near $x$, then a smaller disc compactly contained in that chart. Every germ of a smooth $E$-valued $(0,1)$-form is represented there by $a(z)d\bar{z}\otimes e$; it is $\bar{\mathrm{\partial }}$-closed because there are no $(0,2)$-forms on a curve. By [F4] it has a local primitive after shrinking, so the last map is surjective on stalks. Exactness follows from [F3].

Both ${\mathcal{E}}^0(E)$ and ${\mathcal{E}}^{0,1}(E)$ are sheaves of modules over the real smooth-function sheaf. Apply [F6] on $X$: their positive sheaf cohomology vanishes. The long exact sequence from step 1.1 therefore identifies $H^1(X,{\mathcal{O}}_X(E))$ canonically with the cokernel of the global smooth operator, namely the displayed quotient, by [F12]. Its degree-zero kernel is $H^0(X,{\mathcal{O}}_X(E))=H^0(X,E)$, the holomorphic sections, by [F1]. For a holomorphic bundle map $j:E\to E^{\mathrm{\prime }}$, its local frame coefficient $a$ is holomorphic, so ${\bar{\mathrm{\partial }}}_{E^{\mathrm{\prime }}}(j(fe))=\bar{\mathrm{\partial }}(af)e^{\mathrm{\prime }}=a\bar{\mathrm{\partial }}f{e}^{\mathrm{\prime }}=j({\bar{\mathrm{\partial }}}_E(fe))$. Thus $j$ gives a map of the two Dolbeault resolutions, and naturality of the long exact sequence in [F12] proves naturality of the global identifications. The same long exact sequence gives $H^2(X,{\mathcal{O}}_X(E))=0$ because both degree-one cohomology groups of the smooth terms vanish; in degrees $q>2$ the adjacent higher smooth-term groups vanish as well. By [F2], the smooth operator in the quotient is the restriction of the maximal $L^2$ operator, but no Hilbert-space cokernel identification is used.

Let $W=U_{i_0}\cap \cdots \cap U_{i_r}$ be a nonempty finite intersection. By [F7], $W$ is biholomorphic to a disc, and because it lies in the frame-trivializing member $U_{i_0}$, $E{\mathrm{\mid }}_W$ has a holomorphic frame. In that frame and a disc coordinate, the local Dolbeault quotient is the scalar disc quotient; [F5] makes it zero. Applying the long exact sequence [F12] of the restricted resolution from step 1.1 and the smooth-module acyclicity [F6] shows $H^1(W,{\mathcal{O}}_X(E){\mathrm{\mid }}_W)=0$. The same exact sequence and vanishing of the higher smooth-term cohomology give $H^q(W,{\mathcal{O}}_X(E){\mathrm{\mid }}_W)=0$ for every $q\ge 2$. Thus every nonempty finite intersection is ${\mathcal{O}}_X(E)$-acyclic in the sense of [F8].

The source of ${\phi }_{\mathfrak{U}}^p$ is the fixed-cover group of [F13]. By step 2.2 and the acyclic-cover condition [F8], the cover $\mathfrak{U}$ is Leray; [F9] makes its canonical comparison map an isomorphism in every degree $p\ge 0$ and compatible with refinement. For two allowed covers, compose the first comparison with the inverse of the second; this gives their canonical identification through the same sheaf cohomology group, without requiring a common refinement. Full AC is the choice hypothesis for sheaf cohomology by [F10]; its consequences ${\mathrm{A}\mathrm{C}}_{\omega }$ and DC are used by the smooth-module theorem [F6].

To fix the sign, use the Čech double complex of the acyclic Dolbeault resolution, with the total convention in [F14]. For a smooth splitting $\delta b=c$, holomorphy of $c$ makes ${\bar{\mathrm{\partial }}}_E{b}_i$ agree on overlaps, defining a global $\theta$. In total degree one, $Db=c+\theta$, so $[c]=-[\theta ]$. Thus the unnormalized augmented-resolution identification sends the sheaf comparison class of $c$ to $-[\theta ]$. Multiply that degree-one identification by $-1$ to obtain the normalization stated above; it remains an isomorphism natural in bundle maps. If $b^{\mathrm{\prime }}$ is another splitting, $b_i^{\mathrm{\prime }}-b_i$ glue to a global smooth section, so $[{\bar{\mathrm{\partial }}}_E{b}_i^{\mathrm{\prime }}]=[\theta ]$; refinement compatibility follows from [F9]. This proves the representative rule needed for the residue formula, with no change to the fixed-cover Čech-to-sheaf map.