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BorelSerre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$
The complex structure is natural. The boundary of the locally symmetric space is a subquotient of the boundary of the symmetric space, from which it inherits local properties. As you said, the boundary of 3d hyperbolic space is $\mathbb C\mathbb P^1$; in general, the boundary of hyperbolic space has a unique conformal structure preserved by the group of hyperbolic isometries. 
5h

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Is the Tate conjecture known for etale covers of products of curves
The key point is that the fundamental group of a product is the product of the fundamental groups. This is only true for complete varieties, but given the context of the Tate conjecture the varieties are complete. 
7h

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Detection of stable homotopy by Ktheory spectra
Just the image of J. Nothing can detect more than $K(\mathbb Z)$, because everything factors through that. Moreover, everything remains almost $KU$local. 
18h

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Detection of stable homotopy by Ktheory spectra
$p$adic rings detect everything because they map to characteristic zero fields. I guess my discussion didn't really cover characteristic zero fields or transcendental characteristic $p$ fields, but they detect everything. This reduces to the case of algebraically closed fields. The culmination of Suslin's rigidity argument is the calculation of the $\ell$adic completion of the $K$theory of an algebraically closed field as the $\ell$adic completion of $KU$, with practically no input. Thence the image of $J_U$. 
18h

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Detection of stable homotopy by Ktheory spectra
Yes, the $K$theory of discrete rings is height 1. But the redshift conjecture says that $K$theory raises height. So $K(E_n)$ is supposed to be height $n+1$ (ie, the localization is supposed to be an isomorphism in high degree). Somehow all discrete rings count as height 0, even in positive characteristic. 
18h

answered  Detection of stable homotopy by Ktheory spectra 
Jul 20 
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Are there any useful Grothendieck topologies for which the H1 of $GL_n$ is not the set of rank $n$ vector bundles
I think the cdh cohomology is the same for a scheme and its reduced subscheme, but they do not have the same vector bundles. In characteristic zero, the answer is probably the vector bundles on the reduced subscheme. 
Jul 18 
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Are there any useful Grothendieck topologies for which the H1 of $GL_n$ is not the set of rank $n$ vector bundles
The Nisnevich or cd topology lies between Zariski and etale, so it works. In Voevodsky's cdh topology the structure sheaf and $GL_n$ are not sheaves. But you could still ask about their cohomologies (ie, of their sheafifications). 
Jul 15 
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History of the Proj construction in algebraic geometry
Maybe Serre was the first to go from graded rings to projective varieties, but going from projective varieties to graded rings was quite popular. 
Jul 13 
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What is the lowestweight noncyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?
Linear in $g$ is the range of dimensions where $\overline{\mathcal M}_{g,n}$ has the same cohomology as ${\mathcal M}_{g,n}$, as is the range of dimensions that is generated by tautological classes. So, yes, $i$ must increase with $g$. 
Jul 2 
awarded  Curious 
Jul 1 
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What is the determinant of Poincare duality?
Since Poincare duality generalizes to noncompact manifolds (or manifolds with boundary), you could generalize your question to that context. But now with more spaces, you have MayerVietoris. I think it gives enough additivity to reduce to the case of simplices, which is trivial. 
Jun 27 
awarded  Organizer 
Jun 27 
revised 
A special case of the integer Hodge conjecture
edited tags 
Jun 27 
answered  A special case of the integer Hodge conjecture 
Jun 24 
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Topological razors (balllike spaces)
The last axioms you have to impose is that the Quinn invariant is trivial. There are spaces that from the point of view of point set topology are awfully close to manifolds, but aren't, including ones homotopy equivalent to spheres. 
Jun 24 
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Compact Lie groups with only 3 dimensional cohomology generators
Claudio meant to say "$SO(3)$ is not simplyconnected, but is doubly covered by $S^3$." 
Jun 15 
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A weak version of Bass' conjecture
Stable is easier than unstable. I believe that there are examples of fairly simple fg commutative rings $A$ where $H^m(GL_n(A))$ is infinitely generated but for large $H^m(GL_N(A))$ is finitely generated and independent of large $N$. 
Jun 13 
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Complex non stably trivial complex vector bundle with vanishing Chern classes
$\mathbb R\mathbb P^7$ works, but still with $4E$; $8E$ is trivial. The only cohomological difference between $\mathbb R\mathbb P^6$ and $\mathbb R\mathbb P^7$ is a $\mathbb Z$ is degree $7$. But that's an odd degree, so it contributes neither to Chern classes nor to $K$theory. 
Jun 12 
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Derived category of product of complex manifolds
Bad things happen if you don't assume that your varieties are algebraic (Moishezon). 