8 - Projective space – the statements

Summary

The version of GAGA we plan to prove is about projective space, and it is high time that we disclose to the reader what projective space is. The differential geometer would think of projective space ℂℙⁿ as the quotient of ℂⁿ⁺¹ − {0} by the action of ℂ*. Let us remind the reader.

The group ℂ* is the group of non-zero complex numbers under multiplication. The group ℂ* acts on ℂⁿ⁺¹ by the following rule: the action sends the pair (λ, x), where λ ∈ ℂ* and x ∈ ℂⁿ⁺¹, to λ⁻¹x ∈ ℂⁿ⁺¹. The point 0 ∈ ℂⁿ⁺¹ is a fixed point for this action, and therefore ℂ* acts on the complement ℂⁿ⁺¹ − {0} of 0 ∈ ℂⁿ⁺¹. The orbit space is the space of lines through the origin in ℂⁿ⁺¹, and this is what a differential geometer would understand by projective space ℂℙⁿ.

We have become more sophisticated by now. We would expect to define a ringed space (ℙⁿ, O), which if we are lucky should be a scheme of finite type over ℂ, and so that {ℙ}^an = ℂℙⁿ. The object of the next two chapters is to give the construction. The idea is simple enough. The topological space ℂⁿ⁺¹ can most definitely be viewed as X^an for a scheme (X, O_X) of finite type over ℂ; nothing could be easier.