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

The Dolbeault cohomology group isThe 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 $H_{\bar{\partial }}^{r,r}(X)$.

Solved by gpt-5.6-sol high.