# Second cohomology groups of Nakajima quiver varieties

Let $X=M(v,w)$ be a Nakajima quiver variety for a quiver $Q$. Can one calculate the second singular cohomology groups $H^2(X,\mathbb Z)$ or $H^2(X,\mathbb C)$ explicitly, and if not, are there some particular cohomology classes in these groups which one can identify, or a class of quivers where one can calculate these groups?

I know that in type A the cotangent bundle to the flag variety is a Nakajima quiver variety for the corresponding Dynkin quiver of type A, and the second cohomology group for it is calculated in M. McBreen's talk, which can be found here: http://www.northeastern.edu/iloseu/McBreenTalk.pdf but, if possible, I would like to see a different explanation for the answer in this case from the explanation in this talk.

There's a very natural set of cohomology classes in $H^2(X,\mathbb{Z})$; for each node $i$ in your quiver, there's a tautological bundle $\mathcal{V}_i$. The Chern classes $c_1(\mathcal{V}_i)$ give natural elements of $H^2(X,\mathbb{Z})$. These are usually linearly independent, but not always (for example, sometimes $X$ is a point). Attached to $v,w$, there's a weight space of a highest weight representation for the associated Kac-Moody algebra, and these classes will be linearly independent if the weight is in the interior of the weight polytope for the representation.