Holomorphic curvature of Kähler Einstein metrics on generalised flag manifolds

We analyse the holomorphic curvature of Kähler metrics on generalised flag manifolds with respect to the question of strict positivity. The main results are twofold: Firstly, we show that most generalised flag manifolds with second betti number smaller than 3 have positive holomorphic curvature for any Kähler metric. Secondly, using fairly different techniques we obtain that every generalised flag manifold of rank four or less has positive holomorphic curvature with respect to the Kähler-Einstein metric.