# zbMATH — the first resource for mathematics

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
Full Text: