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Some remarks on multiplier ideals and vector bundles

Roberto Paoletti

2003
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Advances in Geometry
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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.
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... anishing 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 n 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 O PE Ã ð1Þ on the projectivised dual PE Ã is very ample. If X is a projective manifold and Z J X an irreducible subvariety, for every integer p d 1 the p-th symbolic power I h pi Z J O Z 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 ). Theorem 1. Let X be an n-dimensional complex projective manifold. Let E be a rank r very ample vector bundle over X. Let Z J X be a codimension e irreducible subvariety. If H 0 ðX ; Sym d E n I hti Z Þ 0 0, then H 0 ðX ; K X n detðEÞ n Sym nþl E n I Z Þ 0 0 as soon as l d de t Â Ã . Set Y ¼ PE Ã ! p X , the projectivised dual, and let O Y ð1Þ be the relative hyperplane bundle on Y . Let A be any divisor in Y , not necessarily e¤ective, such that O Y ðAÞ ¼ O Y ð1Þ. If D ¼ P i a i D i A Div Q ðX Þ, with the D i H X irreducible divisors, the pull-back of D to Y is the Q-divisor p Ã ðDÞ ¼ P i a i p Ã ðDÞ. We shall say that EðÀDÞ is nef and

doi:10.1515/advg.2003.001
fatcat:46cglphvtfdg7giwjocflkf6pe