# cohomology classes of complex submanifolds

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?

• Your question is ambiguous. Are you asking about properties of cohomology classes of complex subvarieties of $(\mathbb{C}/\mathbb{Z}[i])^2$, in which case the answer is well-known (Lefschetz theorem), or of classes of submanifolds which are complex subvarieties of $T^4$ for some complex structure of $T^4$? (also well-known, but might be more interesting in general).
– abx
Apr 26 '20 at 6:33
• en.wikipedia.org/wiki/Hodge_conjecture Apr 26 '20 at 6:52
• This is related to the Steenrod realizability problem, though that's for the laxer question of representing classes by oriented manifolds. Apr 26 '20 at 18:53
• @abx I just meant cohomology classes of complex subvarieties of $(\mathbb{C}/\mathbb{Z}[i])^2$. Thanks for letting me know about the condition that it has to be in $H^{1,1}(T^4,\mathbb{C}) \cap H^2(T^4,\mathbb{Z})$. I'm sure I'm being very stupid, though, but is it really true that any element in the subgroup generated by complex subvarieties can itself be represented by a complex subvariety? I can see that if C_1 and C_2 are subvarieties, then [C_1] + [C_2] is represented by an actual subvariety, but is the same true of [C_1] - [C_2]? May 4 '20 at 19:14
• No, of course. What I meant is that this is a much studied problem, with proven methods to decide. Look at "divisors and line bundles" in any algebraic geometry book.
– abx
May 4 '20 at 19:40