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A subadditivity property of multiplier ideals. (English) Zbl 1077.14516
Summary: Given an effective \(\mathbb{Q}\)-divisor \(D\) on a smooth complex variety, one can associate to \(D\) its multiplier ideal sheaf \(J(D)\), which measures in a somewhat subtle way the singularities of \(D\). Because of their strong vanishing properties, these ideals have come to play an increasingly important role in higher dimensional geometry. We prove that for two effective \(\mathbb{Q}\)-divisors \(D\) and \(E\), one has the “subadditivity” relation: \(J(D + E) \subseteq J(D) . J(E)\). We also establish several natural variants, including the analogous statement for the analytic multiplier ideals associated to plurisubharmonic functions.
As an application, we give a new proof of a theorem of T. Fujita [Kodai Math. J. 17, No. 1, 1–3 (1994; Zbl 0814.14006)] concerning the volume of a big linear series on a projective variety. The first section of the paper contains an overview of the construction and basic properties of multiplier ideals from an algebro-geometric perspective, as well as a discussion of the relation between some asymptotic algebraic constructions and their analytic counterparts.

MSC:
14E99 Birational geometry
14J17 Singularities of surfaces or higher-dimensional varieties
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