2019/10/30 by Das, Biswarup, Buachalla, Réamonn Ó, Somberg, Petr
#Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.1910.14007
Noncommutative Kähler structures were recently introduced 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 Kähler space which gives a natural set of compatibility conditions between covariant Kähler structures and Woronowicz's theory of compact quantum groups. Each such object admits a Hilbert space completion possessing a remarkably rich yet tractable structure. The analytic behaviour of the associated Dolbeault-Dirac operators is moulded by the complex geometry of the underlying calculus. In particular, twisting the Dolbeault-Dirac operator by a negative Hermitian holomorphic module is shown to give a Fredholm operator if and only if the top anti-holomorphic cohomology group is finite-dimensional. In this case, the operator's index coincides with the twisted holomorphic Euler characteristic of the underlying noncommutative complex structure. The irreducible quantum flag manifolds, endowed with their Heckenberger-Kolb calculi, are presented as motivating examples.