I was wondering if there were restrictions in what the cohomology classes corresponding to complex submanifolds of a complex manifold could be.

For example, say $T^4$ is regarded as a complex manifold as $\mathbb{C}^2/\mathbb{Z}[i] \times \mathbb{Z}[i]$, we have $H^2(T^4) = \Lambda^2(\mathbb{Z}^4) =\mathbb{Z}^6$. If $C$ is a ($1$ complex dimensional) complex submanifold, and we let $[C] \in H^2(T^4)$ be the corresponding cohomology class, could $[C]$ be any cohomology class in $H^2(T^4)$, or are there some restrictions we could impose just by knowing that it is a complex submanifold?

somecomplex structure of $T^4$? (also well-known, but might be more interesting in general). $\endgroup$ – abx Apr 26 at 6:33