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Building on the theory of noncommutative complex structures, the notion of anoncommutative K\"ahler structure is introduced. In the quantum homogeneousspace case many of the fundamental results of classical K\"ahler geometry areshown to follow from the existence of such a structure, allowing for thedefinition of noncommutative Lefschetz, Hodge, K\"ahler-Dirac, and Laplaceoperators. Quantum projective space, endowed with its Heckenberger-Kolbcalculus, is taken as the motivating example. The general theory is then usedto show that the calculus has cohomology groups of at least classicaldimension.

Noncommutative K\"ahler Structures on Quantum Homogeneous Spaces