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Register a new userEquivariant Intersection Cohomology of Toric Varieties
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Authors
Gottfried Barthel
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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.
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We study the multiplication and comultiplication in equivariant cohomology of Sato Grassmannian
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 $mathbb{P} ^N$.
In this paper we construct an additive basis for the cohomology ring of a regular nilpotent Hessenberg variety which is obtained by extending all Poincare duals of smaller regular nilpotent Hessenberg varieties. In particular, all of the Poincare duals of smaller regular nilpotent Hessenberg varieties in the given regular nilpotent Hessenberg variety are linearly independent.
Let $n$ be a fixed positive integer and $h: {1,2,ldots,n} rightarrow {1,2,ldots,n}$ a Hessenberg function. The main results of this paper are twofold. First, we give a systematic method, depending in a simple manner on the Hessenberg function $h$, for producing an explicit presentation by generators and relations of the cohomology ring $H^ast(Hess(mathsf{N},h))$ with $mathbb{Q}$ coefficients of the corresponding regular nilpotent Hessenberg variety $Hess(mathsf{N},h)$. Our result generalizes known results in special cases such as the Peterson variety and also allows us to answer a question posed by Mbirika and Tymoczko. Moreover, our list of generators in fact forms a regular sequence, allowing us to use techniques from commutative algebra in our arguments. Our second main result gives an isomorphism between the cohomology ring $H^*(Hess(mathsf{N},h))$ of the regular nilpotent Hessenberg variety and the $S_n$-invariant subring $H^*(Hess(mathsf{S},h))^{S_n}$ of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the $S_n$-action on $H^*(Hess(mathsf{S},h))$ defined by Tymoczko). Our second main result implies that $mathrm{dim}_{mathbb{Q}} H^k(Hess(mathsf{N},h)) = mathrm{dim}_{mathbb{Q}} H^k(Hess(mathsf{S},h))^{S_n}$ for all $k$ and hence partially proves the Shareshian-Wachs conjecture in combinatorics, which is in turn related to the well-known Stanley-Stembridge conjecture. A proof of the full Shareshian-Wachs conjecture was recently given by Brosnan and Chow, but in our special case, our methods yield a stronger result (i.e. an isomorphism of rings) by more elementary considerations. This paper provides detailed proofs of results we recorded previously in a research announcement.
We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.
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