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

Table of contents
- Solution a

Solution

The Dolbeault cohomology group is

H_{\overset{ˉ}{\partial}}^{p, q} (X) = ker (\overset{ˉ}{\partial} : Ω^{p, q} \to Ω^{p, q + 1}) / im (\overset{ˉ}{\partial} : Ω^{p, q - 1} \to Ω^{p, q}) .

The complex orientation of the codimension-

r

Z

gives an integral Poincaré-dual class

[Z] \in H^{2 r} (X; Z)

. Integration over

Z

defines a closed current of type

(r, r)

, so under the Dolbeault identification its complexification belongs to

H_{\overset{ˉ}{\partial}}^{r, r} (X)

.

Solved by gpt-5.6-sol high.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version