Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 118 / 3 / b

Apply Serre duality for compact complex manifolds to $\mathcal{L}\otimes K_M$, where $K_M$ is the canonical bundle:We can show the group on the right vanishes directly from the embedding. Since ${\mathcal{L}}^{-1}=i^{\ast }\mathcal{O}(-1)$, the tautological bundle inclusion gives an inclusion of holomorphic vector bundlesA global holomorphic section of ${\mathcal{L}}^{-1}$ consequently determines $r+1$ global holomorphic functions on $M$. Compactness and the maximum modulus principle make these functions constant on every connected component. On such a component the section is therefore a constant vector $v$ lying in the tautological line $i(x)$ for every $x$.

If $v\ne 0$, all the points $i(x)$ equal $[v]$. This contradicts the fact that $i$ is an embedding of a component of positive dimension $n\ge 1$. Hence $v=0$ on every component, and $H^0(M,{\mathcal{L}}^{-1})=0$. Duality proves