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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.
We give a complete description of the equivariant quantum cohomology ring of any smooth hypertoric variety, and find a mirror formula for the quantum differential equation.
We investigate the equivariant intersection cohomology of a toric variety. Considering the defining fan of the variety as a finite topological space with the subfans being the open sets (that corresponds to the toric topology given by the invariant o
Let $X^{2n}subseteq mathbb{P} ^N$ be a smooth projective variety. Consider the intersection cohomology complex of the local system $R^{2n-1}pi{_*}mathbb{Q}$, where $pi$ denotes the projection from the universal hyperplane family of $X^{2n}$ to ${(mat
Given a smooth quasiprojective variety $Y$ over $mathbb C$ that is not projective, consider its unordered configuration spaces $mathrm{Conf}^n(Y)$ for $ngeq 0$. Remove a point $P$ of $Y$ and obtain a one-puncture $Y-P$ of $Y$. We give a decomposition
This is an expository lecture, for the Abel bicentennial (Oslo, 2002), describing some recent work on the (small) quantum cohomology ring of Grassmannians and other homogeneous varieties.