Some remarks on multiplier ideals and vector bundles

In this paper, we give two applications of the theory of multiplier ideals to vector bundles over complex projective manifolds, generalizing to higher rank results already established for line bundles. The first addresses the existence of sections of (suitable twists) of symmetric powers of a very ample vector bundle, vanishing on a given subvariety. The second is a vanishing theorem of Gri‰ths type, adjusted with an additional multiplier ideal term as in the standard Nadel vanishing theorem. To begin with, we recall that, starting with work of Bombieri and Skoda, there has been considerable interest in going from hypersurfaces in P highly singular at a given set of points S to hypersurfaces through S of relatively low degree; there is now a very direct and terse approach based on multiplier ideals [3], [4], [5], [6]. Here, the same method is shown to yield a statement in the same spirit about sections of very ample vector bundles on a projective manifold. One says that a rank r holomorphic vector bundle on a complex projective manifold is very ample if the relative hyperplane line bundle OPE ð1Þ on the projectivised dual PE is very ample. If X is a projective manifold and ZJX an irreducible subvariety, for every integer pd 1 the p-th symbolic power I hpi Z JOZ of the ideal sheaf of Z is the ideal sheaf of the holomorphic functions vanishing with multiplicity dp along Z (that is, at a generic point of Z ).