I know Serre gave a way to convert a nice enough scheme over $\mathbb C$ to a complex manifold in his famous GAGA article but is there a way to go back? Here are a few thoughts.
Let $M$ be a noncompact complex manifold, then we can define a locally (maybe not locally, I don't know?) ringed space X to be the set of points on $M$ with the topology defined by letting the sets $f^{-1}(0)$ be closed for all holomorphic functions $ f: M \rightarrow \mathbb C$ and structure sheaf given by $$\mathscr O_X(U) = \mathscr O_M(U)$$
Letting $\mathscr O_M$ be the sheaf $\mathbb C$-valued holomorphic function on M.
Is X a scheme? If so, M obviously cant be compact since then any holomorphic function to $\mathbb C$ is constant but are there other constraints or does all such M define a scheme?
Or would it perhaps be wiser to follow the construction made on p.78 in Hartshorne. Consider t(M) to be the set of all nonempty closed complex submanifolds of M (M is now possiby compact).
The resulting locally (again not sure if its local) ringed space has topology given by defining closed sets to be of the form $t(N')$ for all closed submanifolds N' of M.
Then there is a continuous function $$\alpha : M \rightarrow X, p \mapsto cl\{P\}$$and we let the structure sheaf on t(M) be the sheaf $\alpha_* \mathscr O_M$.
Is t(M) a scheme?
Edit: I just realized that Chows theorem is important for the case where M is a closed submanifold of some $\mathbb{CP}^n$. I think it shows that for such M, t(M) is indeed a scheme.
Second edit: Someone has identified a possible duplicate of my question but this similar question does not answer my very specific question of is X or t(M) schemes for complex manifolds M?