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Let A be a complex torus of (complex) dimension 2 and X the associated Kummer variety A/σ, where σ(x)=x. I would like to compute the cohomology of X with Z coefficients. My initial instinct was to use Mayer-Vietoris, but the exact sequence involves the cohomology of the quadratic cone minus a point which is also proving to be difficult for me. My hope is that as in the case with Q coefficients H1(X,Z)=H3(X,Z)=0,H0(X,Z)=H4(X,Z)=Z and H2(X,Z)=2H1(A) Any tips as to how to compute Hi(X,Z) or, equivalently, places to find tips in the literature would be very helpful. Thank you.

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  • By the Kummer variety do you mean the singular quotient or its resolution once the 16 singular points are blown up? Commented Jun 18, 2010 at 14:13
  • I mean the singular quotient. I suppose I really should have said singular Kummer surface in the title.
    – AJ Stewart
    Commented Jun 18, 2010 at 15:12
  • In that case, why is not just the invariant part of the cohomology? (Perhaps I'm missing something obvious, though.) Commented Jun 18, 2010 at 16:22
  • I edited the title to better reflect the question, by the way. Commented Jun 18, 2010 at 16:38
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    There is 2-torsion; the third mod 2 homology group is nontrivial. Quick proof: Topologically this is what you get from the product of four circle groups by identifying each element with its inverse. As such, it contains as a retract the analogous quotient space of a product of three circle groups. The latter has nontrivial third mod 2 homology, because it is a 3-manifold except for singularities of codimension 3>1. Commented Jun 18, 2010 at 17:52

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I missed that the question concerned the singular Kummer surface (which I think historically was what was what was called the Kummer surface but our current fixation on non-singularity has changed that) so one needs a few more steps than Barth, Peters, van de Ven: Compact complex surfaces (which will be my reference below).

Let π:˜XX be the minimal resolution of singularities and consider the Leray spectral sequence for π. We have πZ=Z and R2πZ the skyscraper sheaf with one Z at each of the 16 singular points. The Leray s.s. thus gives that Hi(X,Z)=Hi(˜X,Z) for i2,3 and hence Hi(X,Z)=Z for i=0,4 and H1(X,Z)=0 as well as a short exact sequence 0H2(X,Z)H2(˜X,Z)vVZvH3(X,Z)0, where V is the set of singular points. Now, it is easy to see that H2(˜X,Z)Zv is given by fdeg(fEv), where Ev:=π1(v). We have deg(fEv)=ev,f, where evH2(˜X,Z) is the fundamental class of Ev. Hence, we get to begin with that H2(X,Z) is the orthogonal complement in H2(˜X,Z) of the ev. By Cor. 5.6 (of BPV) this can be identified with H2(A,Z). On the other hand, the image of H2(˜X,Z) in vVZv contains the linear functions given by the ev and ev(v)=2δv,v so that we may consider the image of H2(˜X,Z) in vVZ/2v. By the fact that the cup product pairing on H2(˜X,Z) is perfect (by Poincaré duality) and by Prop. 5.5 we get that this image is dual to the subspace of affine functions of vVZ/2v (where V is identified by the kernel of multiplication by 2 in A) and hence we get an identification of H3(X,Z) with the dual of the Z/2-space of affine functions of V, in particular it has dimension 5.

Remark: It is interesting to note that while the quotient A/σ as a topological space does not use the complex structure of A it still seems easier to use it (in a very weak form, the blowing up only uses that a conical neighbourhood has a certain form) as we consider the complex blow up of the singular points. Indeed, the use of Mayer-Vietoris tried by the poser does look more difficult (of course that would also use the local form of the singularity but somehow in a less complex fashion).

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