Vector bundles on projective varieties whose restriction to ample\n subvarieties split

We systematically study the splitting of vector bundles on a smooth,\nprojective variety, whose restriction to the zero locus of a regular section of\nan ample vector bundle splits.\n First, we find ampleness and genericity conditions which ensure that the\nsplitting of the vector bundle along the subvariety implies its global\nsplitting. Second, we obtain a simple splitting criterion for vector bundles on\nthe Grassmannian and on partial flag varieties.\n