Abstract: The notion of a covariant Hermitian structure was recently introduced as an algebraic framework in which to perform noncommutative Hermitian geometry on quantum homogeneous spaces. Combining covariant Hermitian structures with Woronowicz's theory of compact quantum groups produces a canonical Hilbert completion carrying a beautiful interaction of analysis, geometry, and algebra. We highlight two aspects of this completion: Firstly, the associated $*$-algebra of smooth functions, and secondly the interaction of Dirac operator index theory with noncommutative Dolbeault cohomology and noncommutative Fano structures. Time permitting, we will discuss the relationship of these structures with Connes' notion of a spectral triple. Throughout, the irreducible quantum flag manifolds, endowed with their Heckenberger--Kolb differential calculus, are presented as motivating examples, focusing in particular on the quantum Grassmannians.