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Authors
Ulrich Bunke
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We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion and define the associated push-forward which satisfies functoriality, compatibility with pull-back diagrams, and projection and bordism formulas. We construct a multiplicative lift of the Chern character from smooth K-theory to smooth rational cohomology and verify that the cohomological version of the Atiyah-Singer index theorem for families lifts to smooth cohomology.
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Let $R$ be the homogeneous coordinate ring of a smooth projective variety $X$ over a field $k$ of characteristic~0. We calculate the $K$-theory of $R$ in terms of the geometry of the projective embedding of $X$. In particular, if $X$ is a curve then we calculate $K_0(R)$ and $K_1(R)$, and prove that $K_{-1}(R)=oplus H^1(C,cO(n))$. The formula for $K_0(R)$ involves the Zariski cohomology of twisted Kahler differentials on the variety.
We give a $K$-theoretic criterion for a quasi-projective variety to be smooth. If $mathbb{L}$ is a line bundle corresponding to an ample invertible sheaf on $X$, it suffices that $K_q(X) = K_q(mathbb{L})$ for all $qledim(X)+1$.
In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory of compact simple Lie groups.
The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe. Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary. These are of independent interest and form the second main theme of the paper. The main technical novelty is the use of large scale index theory for Dirac type operators that are perturbed by lower order operators.
We construct a semi-orthogonal decomposition on the category of perfect complexes on the blow-up of a derived Artin stack in a quasi-smooth centre. This gives a generalization of Thomasons blow-up formula in algebraic K-theory to derived stacks. We also provide a new criterion for descent in Voevodskys cdh topology, which we use to give a direct proof of Cisinskis theorem that Weibels homotopy invariant K-theory satisfies cdh descent.
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