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Register a new userPoisson-de Rham homology of hypertoric varieties and nilpotent cones
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We prove a conjecture of Etingof and the second author for hypertoric varieties, that the Poisson-de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that this conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson-de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2-variable Poisson-de Rham-Poincare polynomial, and prove that it is equal to a specialization of an enrichment of the Tutte polynomial of a matroid that was introduced by Denham. We also compute this polynomial for S3-varieties of type A in terms of Kostka polynomials, modulo a previous conjecture of the first author, and we give a conjectural answer for nilpotent cones in arbitrary type, which we prove in rank less than or equal to 2.
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We use the Beilinson $t$-structure on filtered complexes and the Hochschild-Kostant-Rosenberg theorem to construct filtrations on the negative cyclic and periodic cyclic homologies of a scheme $X$ with graded pieces given by the Hodge-completion of the derived de Rham cohomology of $X$. Such filtrations have previously been constructed by Loday in characteristic zero and by Bhatt-Morrow-Scholze for $p$-complete negative cyclic and periodic cyclic homology in the quasisyntomic case.
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Let $mathcal{O}$ be a Richardson nilpotent orbit in a simple Lie algebra $mathfrak{g}$ over $mathbb C$, induced from a Levi subalgebra whose simple roots are orthogonal short roots. The main result of the paper is a description of a minimal set of generators of the ideal defining $overline{ mathcal{O}}$ in $S mathfrak{g}^*$. In such cases, the ideal is generated by bases of at most two copies of the representation whose highest weight is the dominant short root, along with some fundamental invariants. This extends Broers result for the subregular nilpotent orbit. Along the way we give another proof of Broers result that $overline{ mathcal{O}}$ is normal. We also prove a result connecting a property of invariants related to flat bases to the question of when one copy of the adjoint representation is in the ideal in $S mathfrak{g}^*$ generated by another copy of the adjoint representation and the fundamental invariants.
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