Dolbeault-Dirac Fredholm Operators for Quantum Homogeneous Spaces

Abstract

Noncommutative Hermitian structures were recently introduced in [66] 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 Woronowicz's 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 operator's index coincides with the holomorphic Euler characteristic of the underlying noncommutative complex structure. Our motivating family of examples, the irreducible quantum flag manifolds Oq(G/LS) endowed with their Heckenberger-Kolb calculi, are presented in detail. The noncommutative Bott-Borel-Weil theorem [22] is used to produce a family of Dolbeault-Dirac Fredholm operators for each Oq(G/LS). Moreover, following the spectral calculations of [18], the Dolbeault-Dirac operator of quantum projective space is exhibited as a spectral triple in the sense of Connes.