Let $\pi$ be a finitely generated group. Its character variety is defined as $$Ch(\pi) := \operatorname{Hom}(\pi ,\mathbb C^*) .$$
Why is this a variety? This can be generalised by replacing $\mathbb C^*$, but I think it would be helpful for me to understand this example first.
I noted, that $\operatorname{Hom}(\pi ,\mathbb C^*) = \operatorname{Hom}(\pi_{ab} ,\mathbb C^*) = (\mathbb C^*)^n \oplus T$, wtih $T$ the torsion part. But still, why is this a variety? First of all, is it an affine or projective variety, and what are the defining polynomials?