In the last years there have been a renewed interest for a conjecture by Griffiths stated in 1969. The conjecture characterises the positive characteristic forms for positive (in the sense of Griffiths) holomorphic Hermitian vector bundles: those should be the exactly the forms belonging to the positive cone spanned by Schur forms. After recalling the various definitions of positivity for holomorphic Hermitian vector bundles and (p,p)-forms, we shall explain a recent result, obtained in collaboration with my PhD student F. Fagioli, which partially confirms Griffiths' conjecture. The result is obtained as an application of a pointwise, differential-geometric version of a Gysin type formula for the push-forward of the curvature of tautological bundles over the flag bundle.