A real/complex rational atlas on a smooth closed manifold $M$ is an atlas with charts homeomorphic to Euclidean open sets in $\Bbb{R}^n$/$\Bbb{C}^n$ covering $M$ and real/complex rational transition maps. A real/complex rational structure is a maximal collection of compatible mentioned real/complex atlases. Two rational structures are called isomorphic if there is a topological automorphism $\sigma$ of $M$ such that each coordinate representation of $\sigma$ under a rational chart pair (from each structure) is rational. The existence of a rational structure does not imply being algebraic because every complex torus admits one (the transition maps are translations, thus polynomial), so I want to ask about how rigid these structures are.
Q$1$: Are there topological invariants classifying whether a smooth manifold admits a real/complex rational structure? Or on the contrary every smooth manifold admits one?
Q$2$: Obviously a real/complex rational structure belongs to a smooth/holomorphic structure. Consider the converse problem: what is the moduli space of real/complex rational structures compatible with the smooth/holomorphic structure on $M$? Specifically, is it true that the number of complex rational structures belonging to the holomorphic structure of $M$ is at most $1$? Counterexamples are welcome.
Q$3$: A related question exists on this website, and the first answer gives a sheaf-sense definition of rational structures. However its conclusion — the algebraic structure of $\Bbb{C}^n$ can be pullbacked onto $M$ — seems to contradict my complex torus example. I hope someone can help.
Best Answer
The answers to your question in the case $n=1$ are well-known. In higher dimensions, the answers are less complete, but something is known.
For example, in the real case when $n=1$, there is only one smooth, connected compact $1$-manifold, the circle, and, for each natural number $k\ge1$, there is a rational structure $\mathcal{R}_k$, which is the rational structure induced on the $k$-fold connected cover of $\mathbb{RP}^1\simeq S^1$ (which is smoothly diffeomorphic to $S^1$, of course). Two rational structures $\mathcal{R}_i$ and $\mathcal{R}_j$ on the circle are isomorphic if and only if $i=j$, and every rational structure on the circle is isomorphic to some $\mathcal{R}_i$.
Meanwhile, in the complex case when $n=1$, specifying a rational structure on a compact Riemann surface of genus $g$ (i.e., a topological surface of genus $g$ with a fixed underlying holomorphic structure) is easily seen to be equivalent to specifying a projective structure on the surface. For $g>1$, it is known that the moduli of projective structures on a given compact Riemann surface is equivalent to the moduli of quadratic holomorphic differentials, a complex vector space of dimension $3g{-}3$. For example, see R. C. Gunning's On uniformization of complex manifolds: The role of connections. In particular, these provide counterexamples sought by the OP.
In higher dimensions the situation is more complicated because the pseudogroup of rational maps with a rational inverse, i.e., the so-called birational pseudogroup, in either $\mathbb{R}^n$ or $\mathbb{C}^n$ is not well-understood when $n>1$.
However, when $n=2$, the question of when a given compact complex surface has a rational structure is well-understood since we have a classification of compact complex surfaces, by the work of Kodaira. Not all such surfaces have a rational structure; for example, a K3 surface does not.