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Register a new userA Dolbeault-Dirac Spectral Triple for Quantum Projective Space
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The notion of a Kahler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite Kahler structure has a canonically associated triple satisfying, up to the compact resolvent condition, Connes axioms for a spectral triple. In this paper we begin the development of a robust framework in which to investigate the compact resolvent condition, and moreover, the general spectral behaviour of covariant Kahler structures. This framework is then applied to quantum projective space endowed with its Heckenberger-Kolb differential calculus. An even spectral triple with non-trivial associated K-homology class is produced, directly q-deforming the Dirac-Dolbeault operator of complex projective space. Finally, the extension of this approach to a certain canonical larger class of compact quantum Hermitian symmetric spaces is discussed in detail.
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Noncommutative Hermitian structures were recently introduced by the second author as an algebraic framework for studying noncommutative complex geometry on quantum homogeneous spaces. In this paper, we introduce the notion of a compact quantum homogeneous Hermitian space, which gives a natural set of compatibility conditions between covariant Hermitian structures and Woronowiczs theory of compact quantum groups. Each such object admits a Hilbert space completion, which possesses a remarkably rich yet tractable structure. The spectral behaviour of the associated Dolbeault-Dirac operators is moulded by the complex geometry of the underlying calculus. In particular, twisting the Dolbeault-Dirac operators by a negative (anti-ample) line bundle is shown to give a Fredholm operator if and only if the top anti-holomorphic cohomology group is finite-dimensional. When this is so, the operators index coincides with the holomorphic Euler characteristic of the underlying noncommutative complex structure. Our motivating family of examples, the irreducible quantum flag manifolds endowed with their Heckenberger-Kolb calculi, are presented in detail. The noncommutative Bott-Borel-Weil theorem is used to produce a family of Dolbeault-Dirac Fredholm operators for each quantum flag. Moreover, following previous spectral calculations of the authors, the Dolbeault-Dirac operator of quantum projective space is exhibited as a spectral triple in the sense of Connes.
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