  In this article we deduce criteria for the splitting and the triviality of
vector bundles, by restricting them to partially ample divisors. This allows to
study the problem of splitting on the total space of fibre bundles. The
statements are illustrated with a number of examples. For products of minuscule
homogeneous varieties, our results allow to test the splitting of vector
bundles by restricting them to products of Schubert 2-planes.
  The triviality criteria obtained inhere are particularly suited to Frobenius
split varieties, whose splitting is defined by a section in the anti-canonical
line bundle. As an application, we prove that a vector bundle on a smooth toric
variety, whose anti-canonical bundle has stable base locus of co-dimension at
least three, is trivial when its restrictions to the invariant divisors are
trivial, with trivializations compatible along the various intersections.
