Moduli space of almost complex structures as an algebro-geometric object

Question

Let $M$ be a closed real-analytic manifold of dimension $2n$. Is it possible to make sense of the moduli space of real-analytic almost complex structures on $M$ as an algebro-geometric object (probably a very non-Noetherian one)? Can this be used to gain a new perspective on, say, Fredholm-regular almost complex structures? I would not think that this would be particularly useful, but I think if this is possible it is worth doing just for the fun of it.

Maybe one terribly non-canonical way to do this is to construct it a subscheme of $\bigcup_{p\in M}\mathrm{Mat}(2n, 2n)$ (choose a basis for the fiber of tangent bundle at some point, choose a connection to produce bases in points nearby, somehow make sense of the real-analyticity condition, and then take the vanishing scheme of the "polynomial" equation $J^2=-\mathrm{Id}$ and hope that this can be made to work globally). Hopefully, somebody else thought of a better way.

Answer 1

Let $M$ be a closed $2n$-dimensional smooth orientable manifold, and let $TM$ be its tangent bundle.

Let $\mathcal{A}(M):=\{J\in C^\infty(M,\mathrm{End}(TM))\ |\ J_x^2=-1,\ \forall x\in M\}$. This is an infinite dimensional smooth manifold in general.

We will say $J,J'\in \mathcal{A}(M)$ are equivalent if there exists a diffeomorphism $f\in \mathrm{Diff}(M)$ such that:

1. $f_*(J)=J'$, and
2. there is a continuous map $p:[0,1]\to \mathrm {Diff} (M)$ such that $p(0)=f$ and $p(1)=\mathrm {Id}_M$.

I think this is essentially the conjugation action of $\mathrm{Aut}(TM)$ described here.

Let $\mathcal{J}(M):=\mathcal{A}(M)/\sim$, with respect to the equivalence relation defined above.

Perhaps this is a candidate for a moduli space of almost complex structures on $M$.

When $M=S$ is a surface ($n=1$), all almost complex structures are complex. In that case, the equivalence relation seems to me to be the usual one for Teichmüller space and so $\mathcal{J}(S)=\mathcal{T}(S);$ see here.

As $\mathcal{T}(S)$ covers the moduli space of algebraic curves (of type $S$) via the action of the mapping class group, one could hope that a similar quotient by the general mapping class group $\mathrm{Diff}(M)/\mathrm{Diff}_0(M)$ on $\mathcal{J}(M)$ results in something algebraic.