In Holomorphic Functions of Several Variables by L. Kaup and B. Kaup, the authors have the following discussion in the opening of section 31, which concerns ringed spaces.
If a Hausdorff space $X$ has the structure of a complex manifold, then one can approach a characterisation of that structure in two different ways:
by defining a complex atlas on $X$, or
By distinguishing, in the sheaf $\mathscr{C}_X$ of continuous functions on $X$, a subsheaf $\mathscr{O}_X$ with the following property: each $x \in X$ admits an open neighbourhood $U \subset X$ and a homeomorphism $\varphi : U \to U' \subseteq \mathbb{C}^n$ that induces an isomorphism between $\mathscr{O}_X \vert_U$ and $\mathscr{O}_n \vert_{U'}$.
Note that $\mathscr{O}_n$ denotes the structure sheaf on $\mathbb{C}^n$.
I have the following questions:
- Can we always find such a subsheaf $\mathscr{O}_X$?
- Is this subsheaf unique with respect to the topological space $X$?
- If the Hausdorff space $X$ has the structure of a complex manifold, why do we choose a subsheaf of $\mathscr{C}_X$ rather than a subsheaf of the sheaf of holomorphic functions on $X$?
No, there are many obstructions to having a complex structure, for example $X$ must be even-dimensional, it must have a smooth structure, and then given this there are various conditions on its characteristic classes that must be satisfied.
No, for example there are already uncountably many different complex structures on the torus $T^2$, given by elliptic curves.
This subsheaf is the sheaf of holomorphic functions on $X$, with respect to the complex structure that's being discussed.