  A new framework for noncommutative complex geometry on quantum homogeneous
spaces is introduced. The main ingredients used are covariant differential
calculi and Takeuchi's categorical equivalence for faithfully flat quantum
homogeneous spaces. A number of basic results are established, producing a
simple set of necessary and sufficient conditions for noncommutative complex
structures to exist. Throughout, the family of quantum projective spaces,
endowed with the Heckenberger--Kolb calculus, is taken as the motivating set of
examples.

Quantum Algebra;Algebraic Geometry