An informal introduction to multiplier ideals. (English) Zbl 1084.14015

Avramov, Luchezar L. (ed.) et al., Trends in commutative algebra. Based on lectures presented at the MSRI introductory workshop on commutative algebra held at the Mathematical Sciences Research Institute, Berkeley, CA, USA, September 9–13, 2002. Cambridge: Cambridge University Press (ISBN 0-521-83195-4/hbk). Mathematical Sciences Research Institute Publications 51, 87-114 (2004).
These notes follow a short course on multiplier ideals given by Lazarsfeld at the MSRI in September 2002. The purpose of the lectures was to give a gentle introduction to the algebraically-oriented local side of the theory of multiplier ideals. Section 1 is the introduction describing the subject matter of lectures. Section 2 contains definitions and examples (log resolution of an ideal, definition of multiplier ideal, two basic properties, analytic construction of multiplier ideals, multiplier ideals and tight closure). In Section 3, the multiplier ideals of monomial ideals are discussed, including the topics of toric divisors and the canonical divisor. In Section 4, invariants arising from multiplier ideals and applications are considered (the log canonical threshold, jumping numbers, jumping length, application to uniform Artin-Rees numbers). Section 5 is devoted to local properties of multiplier ideals (Skoda’s theorem, restriction theorem, subadditivity theorem). In Section 6, asymptotic constructions are discussed (graded systems of ideals, asymptotic multiplier ideals, growth of graded systems, a comparison theorem for symbolic powers).
For the entire collection see [Zbl 1056.13001].


14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14C20 Divisors, linear systems, invertible sheaves
Full Text: arXiv Link