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Register a new userAn algebraic index theorem for Poisson manifolds
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The formality theorem for Hochschild chains of the algebra of functions on a smooth manifold gives us a version of the trace density map from the zeroth Hochschild homology of a deformation quantization algebra to the zeroth Poisson homology. We propose a version of the algebraic index theorem for a Poisson manifold which is based on this trace density map.
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We define even dimensional quantum spheres Sigma_q^2n that generalize to higher dimension the standard quantum two-sphere of Podles and the four-sphere Sigma_q^4 obtained in the quantization of the Hopf bundle. The construction relies on an iterated Poisson double suspension of the standard Podles two-sphere. The Poisson spheres that we get have the same symplectic foliation consisting of a degenerate point and a symplectic plane and, after quantization, have the same C^*-algebraic completion. We investigate their K-homology and K-theory by introducing Fredholm modules and projectors.
We consider a smooth Poisson affine variety with the trivial canonical bundle over complex numbers. For such a variety the deformation quantization algebra A_h enjoys the conditions of the Van den Bergh duality theorem and the corresponding dualizing module is determined by an outer automorphism of A_h intrinsic to A_h. We show how this automorphism can be expressed in terms of the modular class of the corresponding Poisson variety. We also prove that the Van den Bergh dualizing module of the deformation quantization algebra A_h is free if and only if the corresponding Poisson structure is unimodular.
Proofs of Tsygans formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the Atiyah-Patodi-Singer index theorem and the Riemann-Roch-Hirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various people the most general version of Tsygans formality conjecture has not yet been proven. In my thesis I propose Fedosov resolutions for the Hochschild cohomological and homological complexes of the algebra of functions on an arbitrary smooth manifold. Using these resolutions together with Kontsevichs formality quasi-isomorphism for Hochschild cochains of R[[y_1, >..., y_d]] and Shoikhets formality quasi-isomorphism for Hochschild chains of R[[y_1,..., y_d]] I prove Tsygans formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold. The construction of the formality quasi-isomorphism for Hochschild chains is manifestly functorial for isomorphisms of the pairs (M, abla), where M is the manifold and abla is an affine connection on the tangent bundle. In my thesis I apply these results to equivariant quantization, computation of Hochschild homology of quantum algebras and description of traces in deformation quantization.
We prove an index theorem for the quotient module of a monomial ideal. We obtain this result by resolving the monomial ideal by a sequence of Bergman space like essentially normal Hilbert modules.
On a complex symplectic manifold we prove a finiteness result for the global sections of solutions of holonomic DQ-modules in two cases: (a) by assuming that there exists a Poisson compactification (b) in the algebraic case. This extends our previous results in which the symplectic manifold was compact. The main tool is a finiteness theorem for R-constructible sheaves on a real analytic manifold in a non proper situation.
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