  We study holomorphic GL(2) and SL(2) geometries on compact complex manifolds.
We show that a compact K\"ahler manifold of complex even dimension higher than
two admitting a holomorphic GL(2)-geometry is covered by a compact complex
torus. We classify compact K\"ahler-Einstein manifolds and Fano manifolds
bearing holomorphic GL(2)-geometries. Among the compact K\"ahler-Einstein
manifolds we prove that the only examples bearing holomorphic GL(2)-geometry
are those covered by compact complex tori, the three dimensional quadric and
those covered by the three dimensional Lie ball (the non compact dual of the
quadric).
