Welcome to MathOverflow

Question

4

Let $X$ be a scheme defined over $\mathbb{C}$ with an involution $\sigma$. How to get a $X_{\mathbb{R}}$ scheme defined over $\mathbb{R}$ such that $X_{\mathbb{R}}\times_\mathbb{R} \mathbb{C} = X$ ? How are the real algebraic bundle on $X_{\mathbb{R}}$ and complex bundle on $X$ related?

edited Aug 4 2010 at 6:24

Charles Matthews
9,720●12●39

asked Aug 4 2010 at 5:13

Yashica
135●3

2

You need a semi-linear $\sigma$, i.e., it covers complex conjugation on $\mathbb C$. In that case the $\mathbb R$-scheme is obtained by descent theory as the quotient by $X$ under $\sigma$ (though I guess in general this quotient will only be an algebraic space). – Torsten Ekedahl Aug 4 2010 at 7:54

3

As an illustration of Torsten's parenthetical remark, on p. 210 of the book "Neron models" there's a nice example (due to Mumford) of a scheme loc. of finite type over $\mathbf{C}[[t]]$ equipped with descent data relative to $\mathbf{R}[[t]]$ so that the descent is an algebraic space and not a scheme. The same example works using $\mathbf{C}[t]$ and $\mathbf{R}[t]$, so it provides schemes of finite type over $\mathbf{C}$ which descend to non-scheme alg. spaces over $\mathbf{R}$. Unless one assumes $X$ is q-projective, it seems hard to determine when the descent as an alg. space is a scheme. – BCnrd Aug 4 2010 at 11:28

Richard Borcherds · Accepted Answer · 2010-08-04 17:36:37Z UTC

The paper by Atiyah on K-theory and reality Quart. J. Math. Oxford Ser. (2) 17 1966 367--386. MR0206940 discusses the topological analogue of this question, showing how to relate complex vector bundles over a space with vector bundles over the fixed points of an involution. The idea is to define a sort of intermediate K-group KR of vector bundles with involution, and compare it to the two other K-groups.

Real algebraic variety and real algebraic bundle

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.