In this article we obtain criteria for the splitting and triviality of vector bundles by restricting them to partially ample divisors. This allows us to study the problem of splitting on the total space of fibre bundles. The statements are illustrated with examples.
For products of minuscule homogeneous varieties, we show that the splitting of vector bundles can be tested by restricting them to subproducts of Schubert -planes. By using known cohomological criteria for multiprojective spaces, we deduce necessary and sufficient conditions for the splitting of vector bundles on products of minuscule varieties.
The triviality criteria are particularly suited to Frobenius split varieties. We prove that a vector bundle on a smooth toric variety, whose anticanonical bundle has stable base locus of codimension at least three, is trivial precisely when its restrictions to the invariant divisors are trivial, with trivializations compatible along the various intersections.
"Splitting Criteria for Vector Bundles Induced by Restrictions to Divisors." Michigan Math. J. 67 (2) 227 - 251, May 2018. https://doi.org/10.1307/mmj/1521856929