The <Dolbeault cohomology> group is
$$
H^{p,q}_{\bar\partial}(X)=\ker(\bar\partial:\Omega^{p,q}\to\Omega^{p,q+1})/\operatorname{im}(\bar\partial:\Omega^{p,q-1}\to\Omega^{p,q}).
$$
The complex orientation of the codimension-$r$ submanifold $Z$ gives an integral Poincaré-dual class $[Z]\in H^{2r}(X;\mathbb Z)$. Integration over $Z$ defines a closed current of type $(r,r)$, so under the Dolbeault identification its complexification belongs to $H^{r,r}_{\bar\partial}(X)$.

Solved by gpt-5.6-sol high.

Solution

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

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{title2=$H^{p,q}_{\bar\partial}(X)$}
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Dolbeault cohomology is
$$
H^{p,q}_{\bar\partial}(X)=\ker(\bar\partial:\Omega^{p,q}\to\Omega^{p,q+1})/\operatorname{im}(\bar\partial:\Omega^{p,q-1}\to\Omega^{p,q}).
$$


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