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We construct a Hecke-equivariant Chow motive whose realizations equal intersection cohomology of Siegel threefolds with regular algebraic coefficients. As a consequence, we are able to define Grothendieck motives for Siegel modular forms.
We consider singular foliations whose holonomy groupoid may be nicely decomposed using Lie groupoids (of unequal dimension). We construct a -theory group and a natural assembly type morphism to the -theory of the foliation -algebra generalizing to the singular case the Baum–Connes assembly map. This map is shown to be an isomorphism under assumptions of amenability. We examine some simple examples that can be described in this way and make explicit computations of their -theory.
We study the Witt groups of perverse sheaves on a finite-dimensional topologically stratified space with even-dimensional strata. We show that has a canonical decomposition as a direct sum of the Witt groups of shifted local systems on strata. We compare this with another “splitting decomposition” for Witt classes of perverse sheaves obtained inductively from our main new tool, a “splitting relation” which is a generalisation of isotropic reduction.
The Witt groups are identified with the (nontrivial) Balmer–Witt groups of the constructible derived category of sheaves on , and also with the corresponding cobordism groups defined by Youssin.
Our methods are primarily algebraic and apply more widely. The general context in which we work is that of a triangulated category with duality, equipped with a self-dual -structure with noetherian heart, glued from self-dual -structures on a thick subcategory and its quotient.
Let be a commutative ring. For any projective -module of constant rank with a trivialization of its determinant, we define a generalized Vaserstein symbol on the orbit space of the set of epimorphisms under the action of the group of elementary automorphisms of , which maps into the elementary symplectic Witt group. We give criteria for the surjectivity and injectivity of the generalized Vaserstein symbol and deduce that it is an isomorphism if is a regular Noetherian ring of dimension or a regular affine algebra of dimension over a perfect field with and .
Given a compact Hermitian complex space with isolated singular points, we construct a Dolbeault-type Hilbert complex whose cohomology is isomorphic to the cohomology of the structure sheaf. We show that the corresponding K-homology class coincides with the one constructed by Baum, Fulton and MacPherson.
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