1,171 research outputs found
Operational K-theory
We study the operational bivariant theory associated to the covariant theory
of Grothendieck groups of coherent sheaves, and prove that it has many
geometric properties analogous to those of operational Chow theory. This
operational K-theory agrees with Grothendieck groups of vector bundles on
smooth varieties, admits a natural map from the Grothendieck group of perfect
complexes on general varieties, satisfies descent for Chow envelopes, and is
A^1-homotopy invariant. Furthermore, we show that the operational K-theory of a
complete linear variety is dual to the Grothendieck group of coherent sheaves.
As an application, we show that the K-theory of perfect complexes on any
complete toric threefold surjects onto this group. Finally, we identify the
equivariant operational K-theory of an arbitrary toric variety with the ring of
integral piecewise exponential functions on the associated fan.Comment: 38 pages; v2: new exampes in Sections 5 and 7, and an new application
(Theorem 1.4), showing that the natural map from K-theory of perfect
complexes to the dual of the Grothendieck group of coherent sheaves is
surjective for complete toric threefolds; v3: final version published in
Documenta Mat
Twisted equivariant K-theory with complex coefficients
Using a global version of the equivariant Chern character, we describe the
complexified twisted equivariant K-theory of a space with a compact Lie group
action in terms of fixed-point data. We apply this to the case of a compact Lie
group acting on itself by conjugation, and relate the result to the Verlinde
algebra and to the Kac numerator at q=1. Verlinde's formula is also discussed
in this context.Comment: Final version, to appear in Topology. Exposition improved, rational
homotopy calculation completely rewritte
Yang-Mills theory and Tamagawa numbers
Atiyah and Bott used equivariant Morse theory applied to the Yang-Mills
functional to calculate the Betti numbers of moduli spaces of vector bundles
over a Riemann surface, rederiving inductive formulae obtained from an
arithmetic approach which involved the Tamagawa number of SL_n. This article
surveys this link between Yang-Mills theory and Tamagawa numbers, and explains
how methods used over the last three decades to study the singular cohomology
of moduli spaces of bundles on a smooth complex projective curve can be adapted
to the setting of A^1-homotopy theory to study the motivic cohomology of these
moduli spaces.Comment: Accepted for publication in the Bulletin of the London Mathematical
Societ
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