Spacetime as an algebraic surface over $\mathbb{C}$

Question

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 $\mathbb{C}$?

Answer 1

The main problems is that "spacetimes", as understood by physicists, are differentiable manifolds equipped with a "causal structure"; see Techinques of differential topology in relativity. It's pretty difficult to see how to define a pseudo-Riemannian structure over a scheme. That's the main obstruction to complete your program.

The best that can be done is probably by means of twistor theory. Twistor theory defines spacetime notions in terms of projective geometry over $\mathbb{CP}^{3}$. Using twistor variables one can understand many spacetime objects such as classical field theory by means of Cech cohomology on twistor space (see Cohomology of massless fields).

The problem is that twistor spaces are very restrictive; they can only be defined for spacetimes with very simple topologies (and asymptotic properties) or by studying the deformation theory of known twistor spaces as analytic manifolds.