Vector bundles on projective varieties which split along q-ample\n subvarieties

Let Y be a subvariety of a smooth projective variety X, and V a vector bundle\non X. Given that the restriction of V to Y splits into a direct sum of line\nbundles, we ask whether V splits on X.\n I answer this question in affirmative if holds: Y is a q-ample subvariety of\nX (for appropriate q), it admits sufficiently many embedded deformations, and\nis very general within its own deformation space. The result goes beyond the\npreviously known splitting criteria for vector bundles corresponding to\nrestrictions. It allows to treat in a unified way examples arising in totally\ndifferent situations.\n I discuss the particular cases of zero loci of sections in globally generated\nvector bundles, on one hand, and sources of multiplicative group actions\n(corresponding to Bialynicki-Birula decompositions), on the other hand.\nFinally, I elaborate on the symplectic and orthogonal Grassmannians; I prove\nthat the splitting of any vector bundle on them can be read off from the\nrestriction to a low dimensional `sub'-Grassmannian.\n