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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.

14C20 Divisors, linear systems, invertible sheaves
14F18 Multiplier ideals
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
Full Text: Euclid arXiv