Suppose G is a group object in the category of algebraic spaces (over a field, if you like, or even over ℂ if you really want). Is G necessarily a scheme?
My feeling is that the answer is "yes" because an algebraic space group which is not a scheme would be too awesome. Any group homomorphism from such a G to an algebraic group (a scheme group) would have to have infinite kernel since an algebraic space which is quasi-finite over a scheme is itself a scheme. In particular, G would have no faithful representations or faithful actions on projective varieties (probably). There can't be a surjective group homomorphism from an algebraic group H to G, since that would identify G with H/K, which is a scheme. In particular, you cannot put a group structure on any étale cover of G.