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A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...
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Can an abelian variety be represented as the cohomology of some other object?
Perhaps you would like to have some Galois action on the cohomology as well?
Putting some additional restriction on $X$, say, demanding it to be a geometric object (like a scheme) would help, too. …