Noncommutative K\"ahler Structures on Quantum Homogeneous Spaces

Building on the theory of noncommutative complex structures, the notion of a noncommutative K\"ahler structure is introduced. In the quantum homogeneous space case many of the fundamental results of classical K\"ahler geometry are shown to follow from the existence of such a structure, allowing for the definition of noncommutative Lefschetz, Hodge, K\"ahler-Dirac, and Laplace operators. Quantum projective space, endowed with its Heckenberger-Kolb calculus, is taken as the motivating example. The general theory is then used to show that the calculus has cohomology groups of at least classical dimension.