Say that the complex manifolds $X=\Bbb{P^1(C)}$ and $X=\Bbb{C/(Z+iZ)}$ are defined over $K=\Bbb{Q}$ and $K=\Bbb{Q}(i)$ respectively, because there are finitely many charts $\phi_j:U_j\to X$ such that $\phi_l^{-1}\circ \phi_j$ sends $K\cap U_j\to K\cap U_l$ (if $K\subset \Bbb{R}$ then $\Bbb{R}\cap U_j$ must be non-empty)
The (projective) elliptic curve $E:zy^2=x^3+5xz^2+z^3$ (without complex multiplication), which is isomorphic to $\Bbb{C/(Z+\tau Z)}$ for some probably transcendental number $\tau$, has some charts defined over two kind of fields:
- $\Bbb{Q}(\tau)$
- The infinite algebraic extension of $\Bbb{Q}$ obtained by adjoining iteratively $a^{1/2}$ and $f^{-1}(a^2)$ for each $a$ in that field, where $f(T)=T^3+5T+1$.
It raises a few questions, for example
Are there other kind of fields defining $E$ ? Which compact complex manifolds are defined over a finite extension of $\Bbb{Q}$ ? Over an extension of $\Bbb{Q}$ finitely generated as a field ?