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Proving things suspected to be true in anabelian geometry is usually very hard. Maybe it is easier to disprove things suspected to be false?

In particular, I am interested in false generalizations of the main conjecture to higher dimensions.

Is there a pair of non-isomorphic varieties, $X$ and $Y$, which one might naively expect to satisfy an anabelian conjecture, but which in fact have the same etale fundamental group?

Obviously I am not interested in simply-connected varieties, abelian varieties, fibrations with the same fundamental group as their base, and other obvious counterexamples. Thus an example must certainly be of dimension $>1$.

Another question is:

Are there guesses about how exactly to reconstruct the geometry of a variety from its etale fundamental group, that might seem true, but are in fact false?

I am thinking about ideas like the Section Conjecture.

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up vote 10 down vote accepted

Grothendieck, in his letter to Faltings, suggested that the moduli space of abelian varieties (quote: "I would assume that the same should hold for the multiplicities of moduli of polarized abelian varieties") would be anabelian. Ihara and Nakamura showed that this was not the case, as the anabelian recipe for the automorphism group gave the wrong answer, so you can take $X=Y$ in your first question. The section conjecture also fails in this case. I have a preprint on that (on my webpage) which also has the reference to the paper of Ihara and Nakamura.

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The moduli of ppav has the surprising property that $A_g(\mathbb C)$ is aspherical in the transcendental topology, but the etale homotopy type is not aspherical. Equivalently, the cohomology of the fundamental group does not match the cohomology of its profinite completion (in Serre's immortal language, the group is not "good"). One way to see this is the congruence subroup theorem, which shows that the etale fundamental group has huge center. So you might classify this example as surprising by failing the (unspecified) hypotheses, not by failing the conclusion. – Ben Wieland Nov 10 '12 at 18:59
Well it's hard to complain about an example that tricked Grothendieck. – Will Sawin Nov 10 '12 at 20:13

Mochizuki proved that the anabelian conjecture for hyperbolic curves (the hom-form) only needs the maximal pro-$p$ quotient of the fundamental groups, for some prime $p$, which gives a stronger result than originally envisioned by Grothendieck.

Y. Hoshi (a student of Mochizuki) proved that an analogous generalization of the section conjecture is false:

Existence of nongeometric pro-p Galois sections of hyperbolic curves, Publications of the Research Institute for Mathematical Sciences (Publ. Res. Inst. Math. Sci.) 46 (2010), no. 4, 829-848.

You can get this paper from his homepage.

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