Vector bundles as direct images of line bundles

Abstract

Let X be a smooth irreducible projective variety over an algebraically closed field K and E a vector bundle on X. We prove that, if dim X≥1, there exist a smooth irreducible projective variety Z over K, a surjective separable morphism f:Z →X which is finite outside an algebraic subset of codimension ≥3 in X and a line bundle L on X such that the direct image of L by f is isomorphic to E. When X is a curve, we show that Z, f, L can be so chosen that f is finite and the canonical map H¹(Z, O) →H¹(X, End E) is surjective.