Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 139 / 2 / iv / Solution

A connected integral projective variety has only constant regular functions, so $h^0(X,{\mathcal{O}}_X)=1$. By Serre duality and $K_X\sim 0$, $h^2(X,{\mathcal{O}}_X)=h^0(X,{\mathcal{O}}_X(K_X))=1$. The assumed first sheaf cohomology vanishes. Therefore the Euler characteristic of a coherent sheaf isThese hypotheses characterize the K3 surface setting used below.