Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 118 / 4 / d / Solution

The blowup is a compact Kähler surface. It is simply connected, so $b_1=b_3=0$, while $b_0=b_4=1$ and the supplied fact gives $b_2=2$. It has no holomorphic two-forms: blowing up a point does not change $H^{2,0}$, and $h^{2,0}({\mathbb{CP}}^2)=0$. Hodge decomposition and conjugation symmetry therefore giveand

Solved by gpt-5.6-sol high.