  Noncommutative K\"ahler 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 \emph{compact quantum
homogeneous K\"ahler space} which gives a natural set of compatibility
conditions between covariant K\"ahler 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.

Quantum Algebra;Differential Geometry;K-Theory and Homology;Operator Algebras