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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 $mathbb{P} ^N$.
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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.
We prove several results concerning the intersection cohomology and the perverse filtration associated with a Lagrangian fibration of an irreducible symplectic variety. We first show that the perverse numbers only depend on the deformation equivalence class of the ambient variety. Then we compute the border of the perverse diamond, which further yields a complete description of the intersection cohomology of the Lagrangian base and the invariant cohomology classes of the fibers. Lastly, we identify the perverse and Hodge numbers of intersection cohomology when the irreducible symplectic variety admits a symplectic resolution. These results generalize some earlier work by the second and third authors in the nonsingular case.
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.
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.
In this paper we give a geometric characterization of the cones of toric varieties that are complete intersections. In particular, we prove that the class of complete intersection cones is the smallest class of cones which is closed under direct sum and contains all simplex cones. Further, we show that the number of the extreme rays of such a cone, which is less than or equal to $2n-2$, is exactly $2n-2$ if and only if the cone is a bipyramidal cone, where $n>1$ is the dimension of the cone. Finally, we characterize all toric varieties whose associated cones are complete intersection cones.
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