Ulrich subvarieties and the non-existence of low rank Ulrich bundles on complete intersections

We characterize the existence of an Ulrich vector bundle on a variety X PN in terms of the existence of a subvariety satisfying some precise conditions. Then we use this fact to prove that a complete intersection of dimension n 4, which if n=4 is very general and not of type (2, 2), does not carry any Ulrich bundles of rank r 3 unless n=4, r=2 and X is a quadric.