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.)