  After recalling briefly some basic properties of the quantum group $GL_q(2)$,
we study the quantum sphere $S_q^2$, quantum projective space $CP_q(N)$ and
quantum Grassmannians as examples of complex (K\"{a}hler) quantum manifolds.
The differential and integral calculus on these manifolds are discussed. It is
shown that many relations of classical projective geometry generalize to the
quantum case. For the case of the quantum sphere a comparison is made with A.
Connes' method.
