Question

I'd be surprised if somebody proved the inverse Galois problem for $\mathbb{C}(x,y)$, but I wanted to make sure.

Answer 1

Surely the inverse Galois problem is known over $\mathbf{C}(x)$: The Galois group of the maximal extension of $\mathbf{C}(x)$ unramified away from $n+1$ given primes of $\mathbf{C}[x]$ is the free profinite group on $n$ generators. Any finite group $G$ is a quotient of such a group, so there exists a finite Galois extension $L/\mathbf{C}(x)$ with Galois group $G$.

Then $L\otimes_{\mathbf{C}(x)}\mathbf{C}(x,y)$ is a finite Galois extension of $\mathbf{C}(x,y)$ with Galois group $G$.