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Using the description of the category of quasicoherent sheaves on a root stack, we compute the -theory of root stacks via localization methods. We apply our results to the study of equivariant -theory of algebraic varieties under certain conditions.
Let be a semisimple Lie group with discrete series. We use maps defined by orbital integrals to recover group theoretic information about , including information contained in -theory classes not associated to the discrete series. An important tool is a fixed point formula for equivariant indices obtained by the authors in an earlier paper. Applications include a tool to distinguish classes in , the (known) injectivity of Dirac induction, versions of Selberg’s principle in -theory and for matrix coefficients of the discrete series, a Tannaka-type duality, and a way to extract characters of representations from -theory. Finally, we obtain a continuity property near the identity element of of families of maps , parametrised by semisimple elements of , defined by stable orbital integrals. This implies a continuity property for -packets of discrete series characters, which in turn can be used to deduce a (well-known) expression for formal degrees of discrete series representations from Harish-Chandra’s character formula.
We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections, making it possible to give concrete descriptions of their derived categories. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskiĭ and Klyachko, and toric varieties associated to Weyl fans of type . Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly nontoric) varieties over nonclosed fields.
We define a -equivariant real algebraic -theory spectrum , for every -equivariant spectrum equipped with a compatible multiplicative structure. This construction extends the real algebraic -theory of Hesselholt and Madsen for discrete rings, and the Hermitian -theory of Burghelea and Fiedorowicz for simplicial rings. It supports a trace map of -spectra to the real topological Hochschild homology spectrum, which extends the -theoretic trace of Bökstedt, Hsiang and Madsen.
We show that the trace provides a splitting of the real -theory of the spherical group-ring. We use the splitting induced on the geometric fixed points of , which we regard as an -theory of -equivariant ring spectra, to give a purely homotopy theoretic reformulation of the Novikov conjecture on the homotopy invariance of the higher signatures, in terms of the module structure of the rational -theory of the “Burnside group-ring”.
For a product of Severi–Brauer varieties, we conjecture that if the Chow ring of is generated by Chern classes, then the canonical epimorphism from the Chow ring of to the graded ring associated to the coniveau filtration of the Grothendieck ring of is an isomorphism. We show this conjecture is equivalent to the condition that if is a split semisimple algebraic group of type , is a Borel subgroup of and is a standard generic -torsor, then the canonical epimorphism from the Chow ring of to the graded ring associated with the coniveau filtration of the Grothendieck ring of is an isomorphism. In certain cases we verify this conjecture.
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