Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 118 / 2 / d

Evaluation is surjective as a sheaf morphism: near $p$, any prescribed value is supplied by a constant holomorphic function. We therefore have a short exact sequence of sheaveswhere ${\mathcal{I}}_p$ is the point ideal sheaf and ${\mathcal{O}}_X$ the sheaf of holomorphic functions. By part (c), the skyscraper sheaf is flasque, so $H^0(X,{\mathbb{C}}_p)=\mathbb{C}$ and $H^i(X,{\mathbb{C}}_p)=0$ for $i>0$.

On the connected compact elliptic curve, the maximum modulus principle makes every global holomorphic function constant. Consequently $H^0(X,{\mathcal{O}}_X)=\mathbb{C}$, and evaluation on global sections is an isomorphism. Also $H^1(X,{\mathcal{O}}_X)\cong \mathbb{C}$: by Serre duality for compact complex manifolds, its dual is $H^0(X,K_X)$, generated by the nowhere-zero form $dz$ descended from $\mathbb{C}$. Higher cohomology of ${\mathcal{O}}_X$ vanishes by the Dolbeault theorem in complex dimension one.

The long exact sequence in sheaf cohomology thus beginsIt follows that