  In this paper we survey geometric and arithmetic techniques to study the
cohomology of semiprojective hyperkaehler manifolds including toric
hyperkaehler varieties, Nakajima quiver varieties and moduli spaces of Higgs
bundles on Riemann surfaces. The resulting formulae for their Poincare
polynomials are combinatorial and representation theoretical in nature. In
particular we will look at their Betti numbers and will establish some results
and expectations on their asymptotic shape.
