We show that the special case of Serre duality involved in the Borel-Bott-Weil theorem can be formulated and proved in the context of equi-variant Kasparov theory by combining the Atiyah-Singer index theorem and the framework of equivariant correspondences developed in another paper by the first author and Ralf Meyer. The twisted Dolbeault cohomology groups of a flag variety that figure in the ...

We formulate and prove the special case of Serre duality involved in the Borel-Bott-Weil theorem in the language of equivariant Kasparov theory. The method is to combine the Atiyah-Singer index theorem and the framework of equivariant correspondences developed in another paper by the first author and Ralf Meyer. The twisted Dolbeault cohomology groups of the flag variety figuring in the Borel-B...

We present a novel proof of Thomason’s theorem relating Bott inverted algebraic K-theory with finite coefficients and étale cohomology for smooth varieties over algebraically closed ground fields. Our proof involves first introducing a new theory, which we term algebraic K-homology, and proving it satisfies étale descent (with finite coefficients) on the category of normal, Cohen-Macaulay varie...

of the underlying vector bundles on X , (i.e. in which we ignore the hermitian metrics). Then the difference ĉh(E0) + ĉh(E2)− ĉh(E1), is represented by a secondary characteristic class c̃h first introduced by Bott and Chern [1] and defined in general in [2]. These Bott-Chern forms measure the defect in additivity of the Chern forms associated by Chern-Weil theory to the hermitian bundles in the ...

We investigate when two generalized real Bott manifolds of height 2 have isomorphic cohomology rings with Z/2 coefficients and also when they are diffeomorphic. It turns out that cohomology rings with Z/2 coefficients do not distinguish those manifolds up to diffeomorphism in general. This gives a counterexample to the cohomological rigidity problem for real toric manifolds posed in [5]. We als...

In this paper we define a complex adapted connection of type Bott on vertical bundle of a complex Finsler manifold. When this connection is flat we get a vertical vector valued cohomology. The notions are introduced by analogy with the real case for foliations. Finally, using the partial Bott connection we give a characterization of strongly Kähler-Finsler manifolds. 2000 Mathematics Subject Cl...

We consider the T–equivariant cohomology of Bott–Samelson desingularisations of Schubert varieties in the flag manifold of a connected semi–simple complex algebraic group of adjoint type with maximal torus T . We construct a combinatorially pure (in the sense of T. Braden and R. Macpherson) sheaf on the Bruhat graph of the associated Weyl group such that its global sections give the T–equivaria...

We give a simplification of the proof of the Bott periodicity theorem presented by Aguilar and Prieto. These methods are extended to provide a new proof of the real Bott periodicity theorem. The loop spaces of the groups O and U are identified by considering the fibers of explicit quasifibrations with contractible total spaces.  2002 Elsevier Science B.V. All rights reserved. AMS classificatio...

We construct new invariants of quadratic forms over commutative rings, using ideas from Topology. More precisely, we de…ne a hermitian analog of the Bott class with target algebraic K-theory, based on the classi…cation of Cli¤ord modules. These invariants of quadratic forms go beyond the classical invariants de…ned via the Cli¤ord algebra. An appendix by J.-P. Serre, of independent interest, de...