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

For the Calabi-Yau threefold in the stated convention, the Hodge decomposition theorem for compact Kähler manifolds givesIn particular, the Dolbeault theorem yields $H^1(M,{\mathcal{O}}_M)=0$. By Serre duality for compact complex manifolds in dimension three,The canonical bundle is trivial, so the group on the right equals $H^1(M,{\mathcal{O}}_M)=0$. Thus $H^2(M,{\mathcal{O}}_M)=0$, and part (c) gives