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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