First of all I would like to say that I know little about physics so I am very likely to talk nonsense in what follows.
If I am not wrong, spacetime can be defined as a $4$-dimensional differentiable manifold $X$ together with a pseudo-Riemannian metric. Hence, one can study the curvature of $X$, geodesics on $X$, the singularities of the metric, etc. Moreover, all these concepts have a physical interpretation.
On the other hand, a $2$-dimensional $\mathbb{C}$-scheme with some extra hypothesis induces a complex analytic space that gives rise to a $4$-dimensional variety over $\mathbb{R}$.
Taking this into account, it could make sense to define spacetime as a complex algebraic surface $Y$ with some good properties and one could study the most common aspects of this type of surfaces - namely, the singularities of $Y$, some Čech cohomology groups of $Y$, the holomorphic Euler characteristic of $Y$, the self-intersection of the canonical class of $Y$ (if defined), etc. Hopefully, these concepts could also have a physical interpretation.
Maybe there is an obvious reason for this not to be worth studied, but I have not been able to find any information about it on the internet. So, my questions are:
Could considering spacetime as an algebraic surface over $\mathbb{C}$ be of any help?
Has someone considered spacetime as an algebraic surface over $\mathbb{C}$?