Complexifying the cotangent bundle of an <almost complex manifold> gives the $\pm i$ eigenspace decomposition $T^*X\otimes\mathbb C=T^{1,0*}X\oplus T^{0,1*}X$. Define
$$
\bigwedge^{p,q}T^*X=\bigwedge^pT^{1,0*}X\otimes\bigwedge^qT^{0,1*}X.
$$
For a form $\alpha$ of type $(p,q)$, the <Dolbeault operator> is the type-$(p,q+1)$ projection of $d\alpha$:
$$
\bar\partial\alpha=\operatorname{pr}_{p,q+1}(d\alpha).
$$

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Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-118/3/a/solution