The talk will begin by defining noncommutative complex and Kähler structures for differential calculi. A covariant differential calculus for $Cq[F_3]$, the full quantum flag manifold of $Cq[SU(3)]$, will then be introduced, and its complex and Kähler structures classified. It will be shown that the dimension of the calculus and the number of classified structures are the same as in the classical case, and that the calculus restricts to the Heckenberger-Kolb calculus on $Cq[CP^2]$. The ring structure of the cohomology ring of the calculus will then be shown to satisfy a $q$-integer deformation of the classical Schubert calculus rules. (Joint work with Petr Somberg.)