Real part of a complex manifold?

Question

I was studying how elliptic curves and complex tori are equivalent, and it got me thinking if one can define the "real part" of a complex manifold.

The motivation is the following. Take $\Lambda \subset \mathbb{C}$ to be a lattice spanned by $\{1,\tau\}$, and let $\wp(z)$ be the Weierstrass $\wp$-function for $\Lambda$. You get a map $z\mapsto (\wp(z),\wp'(z))$ that maps you to a certain elliptic curve. Now, by restricting to $z\in \mathbb{R}$, plotting with Mathematica shows me that this map parameterizes the real points of an elliptic curve. So it seems to me, at least for elliptic curves $E$, there's some notion of the "real part" of the complex manifold underlying $E$. Also, while the underlying manifold always has the topological type of a torus, varying the complex structure also changes the "real part" in an interesting way.

Note: If $X$ is a complex manifold, the hypothetical real part, $M$, should satisfy $\dim_\mathbb{C}X = \dim_\mathbb{R}M$, so I don't want just the underlying manifold of $X$.

My question: is there a well-studied notion of "real part" of a complex manifold? Are there references studying this idea?

I have some partial ideas. If the complex manifold $X$ in question came from an algebraic variety, then one could consider the $\mathbb{R}$-points of $X$, and that could be a definition of the real part of the manifold. This seems to be what is going on in my elliptic curve example. However, I would feel more satisfied if there was a (non-algebraic) description that worked for general complex manifolds (or Kähler, if necessary).

Answer 1

I don't think there is such a notion (I might be wrong though). One reason is that there are many connected, closed, complex manifolds of complex dimension one, but there is only one connected, closed, real manifold of real dimension one, namely $S^1$.

Answer 2

In general $(\wp(x),\wp'(x))$ won't have real coordinates. When $\tau=m/2+it$ for $m\in\Bbb Z$ and $t\in\Bbb R$ then the elliptic curve has real coefficients and $x\mapsto(\wp(x),\wp'(x))$ will parametrise the identity component of the real points. In general though, the curve won't be defined over $\Bbb C$ and $\wp(x)$ and $\wp'(x)$ won't be real for real $x$.