A Splitting Criterion for Vector Bundles on Higher Dimensional Varieties

We generalize Horrocks’ criterion for the splitting of vector bundles on projective space. We establish an analogous splitting criterion for vector bundles on arbitrary smooth complex projective varieties of dimension ≥ 4, which asserts that a vector bundle E on X splits iff its restriction E|Y to an ample smooth codimension 1 subvariety Y ⊂ X splits.