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A characterization of nef and good divisors by asymptotic multiplier ideals. (English) Zbl 1183.14011
A nef Cartier divisor $$D$$ on a complete normal complex algebraic variety $$X$$ is said to be good (or abundant) if its Iitaka dimension $$\kappa(X,D)$$ coincides with its numerical dimension $$\nu(X,D):= \text{sup} \{\nu \in \mathbb N : D^{\nu}\;\text{is not numerically trivial} \}$$. Now let $$D$$ be a divisor on a smooth proper complex variety $$X$$ with $$\kappa(X,D) \geq 0$$. The author proves that $$D$$ is nef and good if and only if the multiplier ideals of sufficiently high multiples of $$|e(D) \cdot D|$$ are trivial, where $$e(D)$$, the exponent of $$D$$, is the g.c.d. of the semigroup of integers $$N(D)= \{m \geq 0 : |mD| \not= \emptyset\}$$. An analogue of this characterization is proven also in the analytic setting.

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14F18 Multiplier ideals 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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