# Affine Algebraic Sub Variety

A subvariety of an affine algebraic variety $V\subseteq\mathbb{C}^n$ is an affine algebraic variety $W\subseteq\mathbb{C}^n$ that is contained in $V$.

So with respect to this definition, is it true that the set $U(n)$ of all unitary matrices is not an affine algebraic subvariety of $\mathbb{C}^{n^2}$?

One way to see it is to notice that $U(n)$ is a compact subset and that non-finite subvarieties of $\mathbb C^n$ are never compact.