Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 118 / 3 / a

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- Solution a

Solution

Complexifying the cotangent bundle of an almost complex manifold gives the

\pm i

eigenspace decomposition

T^{*} X \otimes C = T^{1, 0 *} X \oplus T^{0, 1 *} X

⋀^{p, q} T^{*} X = ⋀^{p} T^{1, 0 *} X \otimes ⋀^{q} T^{0, 1 *} X .

For a form

α

of type

(p, q)

, the Dolbeault operator is the type-

(p, q + 1)

projection of

d α

:

\overset{ˉ}{\partial} α = pr_{p, q + 1} (d α) .

Solved by gpt-5.6-sol high.

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