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Register a new userCohomology of large semiprojective hyperkaehler varieties
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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.
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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 open subsets), equivariant intersection cohomology provides a sheaf (of graded modules over a sheaf of graded rings) on that fan space. We prove that this sheaf is a minimal extension sheaf, i.e., that it satisfies three relatively simple axioms which are known to characterize such a sheaf up to isomorphism. In the verification of the second of these axioms, a key role is played by equivariantly formal toric varieties, where equivariant and usual (non-equivariant) intersection cohomology determine each other by Kunneth type formulae. Minimal extension sheaves can be constructed in a purely formal way and thus also exist for non-rational fans. As a consequence, we can extend the notion of an equivariantly formal fan even to this general setup. In this way, it will be possible to introduce virtual intersection cohomology for equivariantly formal non-rational fans.
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 ${(mathbb{P} ^N)}^{vee}$. We investigate the cohomology of the intersection cohomology complex $IC(R^{2n-1}pi{_*}mathbb{Q})$ over the points of a Severis variety, parametrizing nodal hypersurfaces, whose nodes impose independent conditions on the very ample linear system giving the embedding in $mathbb{P} ^N$.
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 formula that computes the singular cohomology groups of $mathrm{Conf}^n(Y-P)$ in terms of those of $mathrm{Conf}^m(Y); (0leq mleq n)$, and prove it for several families of examples of $Y$, including the case where $Y$ is obtained from a smooth projective variety by puncturing one or more points. This formula keeps track of the mixed Hodge structures of the cohomology groups as well. This result simultaneously implies a result of Kallel involving Betti numbers and a consequence of a combinatorial property of configuration spaces due to Vakil and Wood. We also obtain intermediate results involving ordered configuration spaces that potentially work for more examples of $Y$.
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.
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