Prescribed principal parts on a compact Riemann surface

For each $U_i$, use its disc coordinate $z_i$ and for every $p\in P\cap U_i$ write a finite Laurent representative of ${\eta }_p$ in $z_i-z_i(p)$. Let $f_i$ be the sum of these finitely many principal-part representatives on $U_i$; it has the prescribed principal parts and is holomorphic on $U_i\setminus P$. On overlaps, the prescribed germs cancel at points of $P$ and both lifts are holomorphic elsewhere. Thus $c_{ij}=f_j-f_i$ is holomorphic, and $(f_k-f_j)+(f_j-f_i)=f_k-f_i$ makes it a cocycle. By [F3], its class maps to $\xi (\mathcal{D})$. Changing the lifts adds a holomorphic cochain $a_i$ and hence the coboundary $a_j-a_i$. For two supplied covers, take their finite ordered union and its combined lifts. Differences across the two covers are also holomorphic, so they form one cocycle on this union. Each original cover refines the union by its member inclusion; [F10] maps both restricted cocycles to the same sheaf class. The union need not be good, and no comparison isomorphism for it is used.

Take $D=0$ in [F5]. The local cochain $c_{ij}=f_j-f_i$ has the residue-pairing form $g_{ij}={\eta }_j-{\eta }_i$ with ${\eta }_i=f_i$ and $s_0=1$. For $\omega \in H^0(X,K_X)$, [F5] therefore gives $B_0(\xi (\mathcal{D}),\omega )={\sum }_{q\in X}{\mathrm{Res}}_q(f_i\omega )$. If $q\in P$, the germ of $f_i$ has principal part ${\eta }_q$, so $f_i-{\tilde{\eta }}_q$ is holomorphic at $q$ and [F9] gives ${\mathrm{Res}}_q(f_i\omega )={\mathrm{Res}}_q({\tilde{\eta }}_q\omega )$. If $q\mathrm{\notin }P$, $f_i$ and $\omega$ are holomorphic at $q$, so the residue is zero. This proves the displayed sum over $P$. Replacing ${\tilde{\eta }}_q$ by another representative adds a holomorphic germ, whose product with $\omega$ has zero residue by [F9]. The sum is finite because $P$ is finite.

If a global meromorphic solution $f$ exists, then $a_i:=f_i-f$ is holomorphic on every $U_i$, since its principal parts cancel at every point. Hence $c_{ij}=a_j-a_i$ is a coboundary and $\xi (\mathcal{D})=0$. Conversely, if $\xi (\mathcal{D})=0$, the comparison in [F3] is an isomorphism, so $c$ is a coboundary on $\mathfrak{U}$: there are holomorphic $a_i$ with $f_j-f_i=a_j-a_i$. Then $f_i-a_i=f_j-a_j$ on each overlap, so these meromorphic functions glue to a global meromorphic $f$ whose principal parts are those of $f_i$, namely $\mathcal{D}$.

Fix an arbitrary $\omega \in H^0(X,K_X)$. If $f$ solves $\mathcal{D}$, then $f\omega$ is a global meromorphic differential whose residues are the summands in step 2.1 at points of $P$ and zero elsewhere; [F8] gives that their sum is zero. Conversely, if every displayed residue sum is zero, step 2.1 says $B_0(\xi (\mathcal{D}),\omega )=0$ for every such $\omega$. The injectivity in [F6] forces $\xi (\mathcal{D})=0$, and step 2.2 supplies a global meromorphic solution. Thus the residue condition is necessary and sufficient.