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Poisson–de Rham homology of hypertoric varieties and nilpotent cones

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Abstract

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–Poincaré 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 (J Algebra 242(1):160–175, 2001). 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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Notes

  1. A precise definition of a conical symplectic resolution is given at the beginning of Sect. 3.

  2. Thanks to the anonymous referee for this observation.

  3. The authors thank Carl Mautner for explaining the following two lemmas and their proofs.

  4. More generally, for any triangulated category with a t-structure, if M is a complex whose cohomology is concentrated in negative degrees and N is a complex whose cohomology is concentrated in nonnegative degrees, then \({\text {Hom}}(M, N) = 0\).

  5. It is not a priori clear that \(L_S\) has regular singularities, though this will follow from Corollary 4.5.

  6. Namikawa works in greater generality, not requiring that X be conical. He also uses an a priori different local system than \(L_S\), notated by \(\mathcal {H}\), and defined only on codimension two leaves. In particular, \(\mathcal {H}\) is defined as a topological local system, unlike \(L_S\). However, it follows from the discussion in [31, §4] that \(L_S\) and \(\mathcal {H}\) have the same monodromy; hence, the underlying topological local system of \(L_S\) is isomorphic to \(\mathcal {H}\). Moreover, under our assumptions, we show that \(L_S\) has regular singularities, so it can be viewed as a topological local system isomorphic to \(\mathcal {H}\).

  7. As explained in [32], the full version of the conjecture [32, 3.4] applied to a slice \(X_S\) with a symplectic dual would imply that \(\dim {\text {HP}}_{0}(X_S) = {\text {rk}}K_S\); thus, the hypothesis of Theorem 4.1 would be satisfied.

  8. More precisely, the elements of the image of this orbit under the Killing form isomorphism \(\mathfrak {sl}_r^* \rightarrow \mathfrak {sl}_r\) have this Jordan decomposition.

  9. In fact, in [14] the conclusion \(M \cong T\) of Theorem 4.1 is proved first, and then, the hypothesis follows.

  10. The factor of 2 is there because \(\mathfrak {h}^*\) sits in degree 2.

  11. In fact, in these cases, the result also follows from Example 8.8 and Proposition 8.12.

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Acknowledgments

We would like to thank G. Denham, P. Etingof, C. Mautner, and V. Ostrik for their help with this project. In particular, we thank Mautner for help with Lemmas 3.4 and 3.5 and Etingof for useful discussions about Proposition 2.1 and Conjecture 8.4, and for helpful comments on an earlier version. We are grateful to G. Lusztig for suggesting the formula of Conjecture 8.4 and for introducing us to the material in Remark 8.10. We would also like to thank the anonymous referee for helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Nicholas Proudfoot

  2. Department of Mathematics, University of Texas, Austin, TX, 78712, USA

    Travis Schedler

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  1. Nicholas Proudfoot
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  2. Travis Schedler
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Correspondence to Travis Schedler.

Additional information

Nicholas Proudfoot is supported by NSF grant DMS-0950383.

Travis Schedler is supported by NSF grant DMS-1406553.

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Proudfoot, N., Schedler, T. Poisson–de Rham homology of hypertoric varieties and nilpotent cones. Sel. Math. New Ser. 23, 179–202 (2017). https://doi.org/10.1007/s00029-016-0232-3

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