the subgroup of unitary matrices $\text{U}(n) \subset GL(n, \mathbb{C})$ is compact and definitely algebraic, with an algebraic group law; on the other hand, it's not abelian. why is this not a contradiction, in light of the following facts?
- every complete variety with an algebraic group law is abelian (see e.g. [Milne, Cor. 1.4])
- completeness for varieties over $\mathbb{C}$ amounts to compactness in the complex-analytic topology (see e.g. here);
- $\text{U}(n)$ is algebraic (determined by a determinant relation), with an algebraic group law (matrix multiplication), and is compact (see e.g. here).
these together should imply that $\text{U}(n)$ is abelian, which is clearly false. my best guess is that something is going wrong with completeness, and that this thing isn't actually complete.
what gives?