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If I got it right, there is a theorem due to M. Artin that on a projective complex manifold any point has a Zariski open neighbourhood which is a $K(\pi, 1)$ space. I have two questions about it.

  1. Is there a proof of this result written in English? (I am really bad at French.)

  2. Is it true that a point has an arbitrarily small neighbourhood with this property (i.e. smaller than a given one)?

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  • $\begingroup$ A path-connected space $X$ with $\pi_i(X)=0$ for all $i\ge 2$ is called aspherical (and is indeed a $K(\pi_1(X),1)$). $\endgroup$ – YCor Jul 20 '17 at 10:47
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    $\begingroup$ Eric Friedlander's book on 'etale homotopy theory is written in English. $\endgroup$ – Jason Starr Jul 20 '17 at 11:49
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    $\begingroup$ I just realized how nice it would have been if M. Artin was Martin Artin. I guess you can't win them all... :P $\endgroup$ – Asaf Karagila Jul 20 '17 at 12:25
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Quick answer.

I. See M. Olsson On Faltings’ method of almost etale extensions, chapter 5. He discusses there a version of this fact over a dvr, but I think you can easily extract what you want, if you really don't want to look at SGA.

II. Yes. One possible precise statement is: a smooth scheme $X$ over a field $K$ of characteristic zero admits a basis for the Zariski topology consisting of affine open subsets $U\subseteq X$ which are $K(\pi, 1)$ for the etale topology. Moreover, if $K=\mathbb{C}$, then each $U(\mathbb{C})$ is a $K(\pi, 1)$ as a topological space, and the fundamental group $\pi_1(U(\mathbb{C}))$ is a good group (in the sense of Serre).

Longer answer.

Since I learned some of this from multiple sources for my thesis, I figured I can try to explain what you need to know, especially since some of it is not explicitly stated in SGA.

Definition. An elementary fibration is a map of schemes $f:X\to S$ which admits a factorization (too lazy to draw a diagram here!) as an open immersion $j:X\to \overline X$ followed by a smooth projective $\overline{f}:\overline{X}\to S$ such that the geometric fibers of $\overline{f}$ are integral curves, and such that there exists a complementary to $j$ closed immersion $i:Y\to \overline{X}$ such that $Y\to S$ is finite etale and surjective.

Simply speaking, an elementary fibration is just a fibration in smooth affine curves.

Definition. Let $k$ be a field. An Artin neighborhood over $k$ is a scheme $U/k$ admitting a chain of elementary fibrations $$ U = U_d\to U_{d-1}\to \ldots \to U_1\to U_0 = {\rm Spec}\, k. $$

Artin's theorem consists of two parts:

  1. If $X$ is smooth over an infinite field $k$, then $X$ can be covered by Zariski open Artin neighborhoods. (This uses Bertini, and works in any characteristic).

  2. If $U$ is an Artin neighborhood over a field $k$ of characteristic zero, then $U$ is a $K(\pi, 1)$ for the etale topology. If moreover $k=\mathbb{C}$, then $U(\mathbb{C})$ with the complex topology is a $K(\pi, 1)$ space and $\pi_1(U(\mathbb{C}))$ is a good group.

To understand (2), let us recall the definitions:

Definition. A connected topological space (resp. a connected qcqs scheme) $X$ with a point (resp. geometric point) $x\in X$ is a $K(\pi, 1)$ if for every local system (resp. every locally constant constructible abelian etale sheaf) $\mathscr{F}$ on $X$, the natural map $$ H^*(\pi_1(X, x), \mathscr{F}_x) \longrightarrow H^*(X, \mathscr{F}) $$ is an isomorphism.

In good cases ($X$ a CW complex or locally noetherian), this is equivalent to the vanishing of $\pi_q(X)$ for $q>1$ (in the etale setting, these are the etale homotopy groups of Artin and Mazur). Examples of $K(\pi, 1)$'s are smooth curves not isomorphic to $\mathbb{P}^1$ or spectra of fields.

Definition. A group $G$ is a good group if for every finite $G$-module $M$, the natural maps $$ H^*(\hat G, M)\longrightarrow H^*(G, M) $$ are isomorphisms (here $\hat G$ denotes the profinite completion of $G$).

Finite groups, free groups, and extensions of good groups are good - this will be important in a second. A nice example of a group which is not good is ${\rm Sp}(2g, \mathbb{Z})$ for $g>2$ (I don't remember the exact bound). This implies that the moduli space of principally polarized complex abelian varieties with suitable level structure is a $K(\pi, 1)$ in the topological sense but not for the etale topology.

To understand why (2) is true, it is enough to know the following:

Lemma. Let $f:X\to S$ be an elementary fibration over a regular noetherian $\mathbb{Q}$-scheme $S$. Then

(a) If $S$ is a $K(\pi, 1)$ then so is $X$,

(b) If $S$ is of finite type over $\mathbb{C}$ and $S(\mathbb{C})$ is a $K(\pi, 1)$, then $X(\mathbb{C})$ is a $K(\pi, 1)$,

(c) If $S$ is of finite type over $\mathbb{C}$, then $\pi_1(X(\mathbb{C}))$ is an extension of $\pi_1(S(\mathbb{C}))$ by a free group. In particular, if $\pi_1(S(\mathbb{C}))$ is good, so is $\pi_1(X(\mathbb{C}))$.

Assertions (b) and (c) follow easily from the (not entirely obvious but intuitively clear) fact that $f:X(\mathbb{C})\to S(\mathbb{C})$ is a locally trivial fibration whose fiber is an open Riemann surface, whose homotopy type is that of a wedge of circles. Assertion (a) can be proved directly using Abhyankar's lemma (this is where we need characteristic zero, otherwise the proof fails due to wild ramification problems) and the Leray spectral sequence, cf. Lemma 5.5 in Olsson's article. Alternatively, assertion (a) can be deduced from (b) and (c) if $S$ is of finite type over $\mathbb{C}$ using the comparison theorems between singular and etale cohomology, but that would be a wrong thing to do since Artin's original motivation was to prove that comparison!

Which brings us to why Artin proved this theorem in the first place.

Application. Let $X$ be smooth and of finite type over $\mathbb{C}$ and let $\mathscr{F}$ be an locally constant constructible sheaf on $X$. Then the natural maps $$ H^*(X, \mathscr{F})\longrightarrow H^*(X(\mathbb{C}), \mathscr{F}) $$ are isomorphisms.

If $X$ is an Artin neighborhood, then this follows from the fact that the above maps can be factored as (sorry, I forgot how to draw a commutative square on MO): $$ H^*(X, \mathscr{F}) \cong H^*(\pi_1(X, x), \mathscr{F}_x) \cong H^*(\pi_1(X(\mathbb{C}), x), \mathscr{F}_x) \cong H^*(X(\mathbb{C}), \mathscr{F}). $$ Here the first isomorphism comes from the fact that $X$ is a $K(\pi, 1)$ as a scheme, the second from the fact that $\pi_1(X(\mathbb{C}),x)$ is a good group (combined with the theorem from SGA1 which says that $\pi_1(X, x)$ is the profinite completion of $\pi_1(X(\mathbb{C}), x)$), and the last one from the fact that $X(\mathbb{C})$ is a $K(\pi, 1)$ space. If $X$ is not necessarily an Artin neighborhood, it can be covered by such, and their pairwise intersections can be covered by such, etc., and on both sides we can write a spectral sequence of a hypercovering.

In fact, this application is sort of a red herring, as later Artin proved a much more general result (with $X$ finite type but not necessarily smooth and $\mathscr{F}$ an arbitrary constructible etale sheaf) using reduction to curves. Still, Artin's result has had some influence on algebraic geometry, for example it showed up in Faltings' work on $p$-adic Hodge theory (cf. Olsson's paper!).

A sketch of Artin's construction (added Aug 6, 2017).

I found the following short explanation of Artin's construction of elementary neighborhoods on my hard drive. It is enough to prove:

Lemma. Let $X$ be a smooth variety over an infinite field $k$, let $x\in X(k)$ be a point. Then there exists a Zariski neighborhood $U$ of $x$ and an elementary fibration $\pi:U\to S$.

Proof (sketch). We can assume $X$ is affine, embed $X$ into a normal projective $\bar X\subseteq \mathbf{P}^N$, let $Y = \bar X - X$ be the complement. By a Bertini-type argument, after passing to a higher Veronese embedding, there exists a linear subspace $L$ of codimension $\dim X - 1$ containing $x$, avoiding the singularities of $\bar X$ and $Y$ and transverse to $\bar X$ and $Y$. We can furthermore find a subspace $C\subseteq L$ of codimension 1, not containing $x$, disjoint from $Y$ and meeting $X\cap L$ transversely. The projection $\mathbf{P}^N\setminus C\to \mathbf{P}^{\dim X - 1}$ along $C$ is a rational map which extends to the blow-up of $\mathbf{P}^N$ along $C$. Taking the proper transform $\bar X'$ of $\bar X$ in this blow-up, we get a proper surjective map $\bar f: \bar X'\to \mathbf{P}^{\dim X - 1}$ whose fiber at $\bar f(x)$ is the smooth curve $L\cap \bar X$. Then $\bar f$ defines an elementary fibration as needed.

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  • $\begingroup$ Thank you. I am more interested in the Bertini part here. (Because I think I know how to handle the complex topology.) And about II (Zariski basis): $\endgroup$ – Alex Gavrilov Jul 20 '17 at 12:43
  • $\begingroup$ is it sufficiently obvious or something besides the actual theorem is necessary? $\endgroup$ – Alex Gavrilov Jul 20 '17 at 12:46
  • $\begingroup$ Is there an easy example where $X$ is $K (\pi, 1)$ and yet it has some nontrivial higher étale homotopy group? $\endgroup$ – user40276 Jul 20 '17 at 20:55
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    $\begingroup$ I think the badness of $Sp_{2g}$ occurs for $g \geq 2$, as that's where the congruence subgroup property is known. $\endgroup$ – Will Sawin Jul 20 '17 at 21:17
  • $\begingroup$ @AlexGavrilov Yes, part II follows directly from Artin's theorem. About the Bertini part, I'm sorry but I would have to translate the relevant part of SGA4 for you, I'm not sure there's a better reference. I encourage you to try reading it (SGA4 part III Expose XI sections 2-3). And as I said I think all the details are contained in Olsson's paper. $\endgroup$ – Piotr Achinger Jul 20 '17 at 21:31

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