The moduli of substructures of a compact complex space

We construct a space $W_X$ of fine moduli for the substructures of an arbitrary compact complex space $X$. A substructure $(X,{\cal A})$ of $X$ is given by a subalgebra ${\cal A}$ of the structure sheaf $\text{${\cal O}_{X}$}$ with the additional feature that $(X,{\cal A})$ is also a complex space; $(X,{\cal A})$ and $(X,{\cal A'})$ are called equivalent if and only if ${\cal A}$ and ${\cal A'}$ are isomorphic as subalgebras of $\text{${\cal O}_{X}$}$.

Since substructures are quotients, it is only natural to start with the fine moduli space $Q_X$ of all complex-analytic quotients of $X$. In order to obtain a representable moduli functor of substructures, we are forced to concentrate on families of quotients which satisfy some flatness condition for relative differential modules of higher order. Considering the corresponding flatification of $Q_X$, we realize that its open subset $W_X$ consisting of all substructures turns out to be a complex space which has the required universal property.