  Using the framework of noncommutative Kahler structures, we generalise to the
noncommutative setting the celebrated vanishing theorem of Kodaira for positive
line bundles. The result is established under the assumption that the
associated Dirac-Dolbeault operator of the line bundle is diagonalisable, an
assumption that is shown to always hold in the quantum homogeneous space case.
The general theory is then applied to the covariant Kahler structure of the
Heckenberger-Kolb calculus of the quantum Grassmannians allowing us to prove a
direct q-deformation of the classical Grassmannian Bott-Borel-Weil theorem for
positive line bundles.
