In the last years there has been a renewed interest around a long-standing conjecture by Griffiths characterizing which should be the positive characteristic forms for any Griffiths positive vector bundle. This conjecture can be interpreted as the differential geometric counterpart of the celebrated Fulton–Lazarsfeld theorem on positive polynomials for ample vector bundles. In this talk, we present some results that confirm the above conjecture for several characteristic forms. The positivity of these forms is due to a theorem which provides the version at the level of representatives of the universal push-forward formula for flag bundles valid in cohomology.