Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 2

For an ordered open cover $\mathcal{U}=(U_i)$ and a sheaf of sets $\mathcal{F}$, the Čech cochain group isIts Čech coboundary isTerms in ${\delta }^2$ cancel in pairs. Hence

Take pairwise disjoint coordinate discs $U_m$ around $x_m$ and put $U_0=S\setminus \{x_1,\dots ,x_N\}$. A prescribed principal part of a meromorphic function $P_m$ is holomorphic on $U_0\cap U_m$. Definewith all other components zero. No three distinct cover members meet, so the Čech cocycle condition is automatic. Thus $(P_{ij})$ is a Čech one-cocycle for $\mathcal{O}$.

If ${\check{H}}^1(S,\mathcal{O})=0$, that cocycle is a Čech coboundary: there are $f_i\in \mathcal{O}(U_i)$ withSet $F=-f_0$ on $U_0$ and $F=P_m-f_m$ on $U_m$. These formulas agree on overlaps, so the sheaf gluing axiom gives a global meromorphic function. Moreover $F-P_m=-f_m$ is holomorphic near $x_m$. Thus $F$ solves the Mittag-Leffler problem on a Riemann surface with precisely the prescribed principal parts.

Let ${\Omega }^p$ be the sheaf of holomorphic $p$-forms and ${\mathcal{A}}^{p,q}$ the sheaf of smooth $(p,q)$-forms. The Dolbeault-Poincaré lemma makesan exact resolution by fine sheaves, which are acyclic for global sections. Its global cochain complex computes Dolbeault cohomology, proving the Dolbeault theorem