# Are there moduli spaces of higher-dimensional varieties

In short, the answer to the question is yes. I'm aware of the existence of moduli spaces for canonically polarized varieties with fixed Hilbert polynomial over $\mathbf C$. I think they require the minimal model program, but that should be fine.

My real question is whether these moduli spaces extend to stacks defined over $\mathbf Z$.

Is there a moduli stack over $\mathbf Z$ of canonically polarized surfaces of general type with fixed Hilbert polynomial?