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

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 $$\mathbb{C}$$?

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).