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I've been thinking recently about moduli spaces defined over $\mathbb{Z}$, and this led me to the following question:


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 $Spec(\mathbb{C})\rightarrow X_{\mathbb{C}}$.


b. It is the set of all sections of the structure morphism $X_{\mathbb{C}}\rightarrow Spec(\mathbb{C})$.


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?


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.

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It should be the second (b) with the analytic topology. In the first case, you would get the union of $\sigma(X(\mathbb{C}))$ as $\sigma$ varies over $Aut(\mathbb{C})$, which would be quite strange –  Donu Arapura Jul 19 '11 at 19:04
Then, if I have a scheme $X$ over $\mathbb{Z}$, what can I say about $X(\mathbb{C})$? Can I say anything constructive? –  Makhalan Duff Jul 19 '11 at 19:07
Let me give a rephrasing of what I'm perplexed about. Say $X$ is the (coarse, fine, it doesn't matter) moduli space over $\mathbb{Z}$ of some moduli problem. Say $W_{\mathbb{C}}$ is the moduli space over $\mathbb{C}$. My intuition says that $X\times Spec(\mathbb{C})\cong W_{\mathbb{C}}$. But $X(\mathbb{C})$ has a different interpretation from $W(\mathbb{C})$? This seems weird, since they need to classify the same thing! –  Makhalan Duff Jul 19 '11 at 19:15
It seems that the main question here was settled. However, I remain confused. So I will start a new question with my new source of confusion (basically summed up in the above comment). –  Makhalan Duff Jul 19 '11 at 19:24
It can be found here: mathoverflow.net/questions/70773/… –  Makhalan Duff Jul 19 '11 at 19:38
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1 Answer

up vote 2 down vote accepted

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 SGA 1, Exp. 12, and in Serre's GAGA.

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.

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