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.



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> 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?
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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:



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> 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?
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I am thinking about ideas like the Section Conjecture.



What conjectures in anabelian geometry are false?
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.


