Let X = complex manifold of dimension n. Thus, it's a real manifold of dimension 2n.
Now cohomology is a topological concept so it should not depend upon the structure given on a topological space.
We know that k'th Singular cohomology of X is 0 for k > 2n. We can also define a sheaf cohomology on that space using derived functor approach of Grothendieck. Then (by a result of Grothendieck) we know that k'th sheaf cohomology is 0 for k > n.
Now, for constant sheaves [say R], the sheaf cohomology agrees with singular cohomology [with coefficient R]. Does this means that even the k'th singular cohomology of X vanishes for k > n ??
[Edited] I now feel that the result which says that sheaf and singular agrees is actually this that k'th sheaf cohomology [of a complex manifold and constant sheaf] will agree with 2k'th singular cohomology [of the underlying real manifold]. Is this correct?? I would still like others to comment.