The hypertoric intersection cohomology ring

نویسنده

  • Tom Braden
چکیده

We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset. A hypertoric variety is a symplectic algebraic variety, equipped with a torus action, whose structure is determined by the geometry and combinatorics of a rational hyperplane arrangement in much the same way that a toric variety is determined by a rational convex polyhedron (or, more generally, a rational fan). Since hypertoric varieties were first introduced by Bielawski and Dancer [BD], many of their algebraic invariants have been computed in terms of the associated arrangements. In particular, combinatorial formulas have been given for the ordinary and equivariant cohomology rings of a smooth hypertoric variety [HS,Ko,HP], and for the intersection cohomology Betti numbers of a (very singular) affine hypertoric variety [PW]. In this paper, we refine the results of [PW] to give a combinatorial computation of the equivariant intersection cohomology groups of a hypertoric variety MH associated to an arbitrary rational hyperplane arrangementH (Theorem 2.7). We use this to prove a conjecture of [PW, 6.4], which states that the intersection cohomology of a hypertoric variety has a natural ring structure (Corollary 4.5). In the special case where H is central and unimodular (which is equivalent to saying that MH is affine and has a hypertoric resolution of singularities), we show that this ring structure exists on the deepest possible level, namely on the equivariant IC sheaf in the equivariant derived category (Theorem 5.1). The unit element in this ring structure is given by the natural map from the constant equivariant sheaf, which implies that our ring structure “behaves like a cup product”. In particular, for example, it implies that the restriction map from the equivariant intersection cohomology of MH to the equivariant cohomology of the generic stratum will be a ring homomorphism. We prove these results using a general notion, which we develop in Section 1, of localization from T -equivariant constructible sheaves on an equivariantly stratified T -space to sheaves on a poset whose elements index the strata, equipped with a linear structure that keeps track of the stabilizer on the associated stratum. In this framework, we identify the total equivariant intersection cohomology group IH T (MH) with the space of sections of a sheaf L, called a minimal extension sheaf, on the poset LH of flats of H. The flats index the strata of a natural stratification of MH, and we show that the stalk of L at any flat is Supported in part by NSF grant DMS-0201823. Supported in part by an NSF Postdoctoral Research Fellowship and NSF grant DMS-0738335.

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تاریخ انتشار 2008