 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-118/4/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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} (CP^{2}) = 0

. Hodge decomposition and conjugation symmetry therefore give

h^{0, 0} = h^{2, 2} = 1, h^{1, 1} = 2,

and

h^{1, 0} = h^{0, 1} = h^{2, 0} = h^{0, 2} = h^{2, 1} = h^{1, 2} = 0.

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