In Riemann Existence, what is the interpretation of the space of complex-geometric points? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T08:21:12Z http://mathoverflow.net/feeds/question/70765 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70765/in-riemann-existence-what-is-the-interpretation-of-the-space-of-complex-geometri In Riemann Existence, what is the interpretation of the space of complex-geometric points? Makhalan Duff 2011-07-19T18:50:39Z 2011-07-19T19:23:42Z <p>I've been thinking recently about moduli spaces defined over $\mathbb{Z}$, and this led me to the following question:</p> <h3>Question</h3> <p>Riemann existence says that if we have a variety over $\mathbb{C}$, $X_{\mathbb{C}}$, then $\widehat{\pi_1^{top}(X_{\mathbb{C}}(\mathbb{C}))}\cong\pi_1^{et}(X_{\mathbb{C}})$. For which (the first, the second, or both) of the following interpretations of $X_{\mathbb{C}}(\mathbb{C})$ does this theorem work: a. It is the set of all morphisms $Spec(\mathbb{C})\rightarrow X_{\mathbb{C}}$.</p> <p>or</p> <p>b. It is the set of all sections of the structure morphism $X_{\mathbb{C}}\rightarrow Spec(\mathbb{C})$.</p> <h3>Motivation</h3> <p>The motivation to the question, as I said before, is because I've been thinking about moduli spaces over $\mathbb{Z}$. Let's say $X$ is a priori a scheme of finite type over $\mathbb{Z}$. Here $X(\mathbb{C})$ has only the first interpretation, since the structure morphism is not going to $\mathbb{C}$, but to $\mathbb{Z}$. I want to say something about the geometry of $X(\mathbb{C})$, but how could I if this is not what Riemann Existence is talking about?</p> <h3>Remark</h3> <p>Let me give you a sense of why the two interpretations are so different. If $X_{\mathbb{C}}$ is a variety over $\mathbb{C}$, the second interpretation would mean that $X_{\mathbb{C}}(\mathbb{C})$ is in correspondence with the maximal points of $X_{\mathbb{C}}$. However, in the first interpretation we would also have phenomena like the following: If $Y_{\mathbb{C}}$ is a subvariety of $X_{\mathbb{C}}$, then $\kappa(Y_{\mathbb{C}})$ (the function field of $Y_{\mathbb{C}}$) is going to be a field of cardinality $2^{\aleph_0}$ and so its algebraic closure is isomorphic to $\mathbb{C}$. This means that we have a $Spec(\mathbb{C})\rightarrow X_{\mathbb{C}}$ such that its image is at the generic point of $Y$. Obviously, the first interpretation makes more geometric sense, but as I mentioned in the motivation, I am very interested in how to deal with the second. And so, in particular, I'm asking if we have a Riemann Existence theorem for the second interpretation.</p> http://mathoverflow.net/questions/70765/in-riemann-existence-what-is-the-interpretation-of-the-space-of-complex-geometri/70769#70769 Answer by S. Carnahan for In Riemann Existence, what is the interpretation of the space of complex-geometric points? S. Carnahan 2011-07-19T19:18:26Z 2011-07-19T19:23:42Z <p>The set of points should be given by the second choice, i.e., the set of $\operatorname{Spec} \mathbb{C}$-points, over $\operatorname{Spec} \mathbb{C}$. However, there is an additional step you need to do before defining $\pi_1$ (besides choosing a basepoint), which is applying an analytification functor to endow the set with a suitable topology. This functor takes locally finite type schemes over $\operatorname{Spec} \mathbb{C}$ to complex analytic spaces. There is a brief exposition of analytification in <a href="http://arxiv.org/abs/math/0206203" rel="nofollow">SGA 1, Exp. 12</a>, and in Serre's GAGA.</p> <p>Regarding your comment about making a topological space from a variety defined over $\mathbb{Z}$, a $\mathbb{C}$-point of the base change to $\mathbb{C}$ over $\mathbb{C}$ is the same as a $\mathbb{C}$-point of the original scheme. This is the universal property of fiber product.</p>