Did you know that multiplier ideals now have an MSC code? In the MSC2010 revision they are given code 14F18. (Up to now, papers on multiplier ideals have been scattered over a number of codes involving singularities, local study, birational geometry, etc.) This is a nice development that will certainly make the multiplier ideals literature easier to deal with in the future.

Some introductions to multiplier ideals

• Blickle--Lazarsfeld, ``An informal introduction to multiplier ideals'', arxiv:0302.5351. This covers some properties including vanishing theorems, Skoda's theorem, and restriction and subadditivity properties; monomial ideals; jumping numbers; asymptotic multiplier ideals; two uniform results in commutative algebra (uniform Artin-Rees numbers and uniform bounds for symbolic powers). This is the place to start if you are wondering ``what is a multiplier ideal, and why are they interesting?''
• Lazarsfeld, Positivity in Algebraic Geometry, volume 2, Springer Ergebnisse. This has the definitive, complete treatment and the most complete bibliography.
• Grushevsky, ``Multiplier ideals in algebraic geometry'', arxiv:0502.5387. This discusses vanishing theorems and general properties; singularities of theta divisors; asymptotic multiplier ideals; Siu's proof of deformation invariance of plurigenera of varieties of general type; multiplier ideals in the analytic setting; analytic proofs of vanishing theorems and invariance of plurigenera.
• Ein, ``Multiplier Ideals, Vanishing Theorem and Applications'', arxiv:9709.5015. This paper surveys both vanishing theorems and multiplier ideals, and discusses several applications, including singularities of theta divisors; adjoint linear series; a theorem of Levine on invariance of plurigenera (following an idea of Siu); a theorem of Esnault and Viehweg on zeros of polynomials. Being over ten years old, it does not always reflect the current notation.
• Demailly, Analytic methods in algebraic geometry, http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/eem2007.pdf. This is a textbook on complex geometry and analytic methods such as L^2 estimates.
• Siu, ``Multiplier ideal sheaves in complex and algebraic geometry'', arxiv:0504259. (From the Arxiv abstract: This article discusses the geometric application of the method of multiplier ideal sheaves. It first briefly describes its application to effective problems in algebraic geometry and then presents and explains its application to the deformational invariance of plurigenera for general compact algebraic manifolds. Finally its application to the conjecture of the finite generation of the canonical ring is explored and the use of complex algebraic geometry in complex Neumann estimates is discussed.) This is not elementary but gives an idea of how multiplier ideals bridge between algebraic geometry and complex geometry.
• Ein--Mustata, ``Invariants of singularities of pairs'', arxiv:0604.5601. An update as of August 2006. Discusses different approaches to multiplier ideals including the definitions by local integrability, by resolution of singularities, by arc spaces, and by characteristic p methods. Discusses a large number of applications, including some very recent ones.
• Lazarsfeld, ``A short course on multiplier ideals'', arxiv:0901.0651.
• Dissertations by, for example, Jason Howald (a short survey on multiplier ideals, computation for monomial ideals), Amanda Johnson (a [different] short survey on multiplier ideals, computation for generic determinantal ideals), myself (a short very elementary discussion of how to compute multiplier ideals, computation for line arrangements), and an increasing number of other students who have written dissertations on multiplier ideals.

In addition you may be interested in these:

• Some expository notes based on talks I gave in the TAMU Several Complex Variables seminar in February 2008 on multiplier ideals, aimed at relating resolution of singularities to the problem of simplifying integrals. (July 16, 2008: v0.2. Numerous minor improvements.)
• A talk I gave at UT Austin in April 2008 on multiplier ideals of hyperplane arrangements.

And here is a list of my publications about multiplier ideals:

1. A note on Mustata's computation of multiplier ideals of hyperplane arrangements, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1575--1579; Link to journal, Link to arXiv
2. Multiplier ideals of general line arrangements in C^3, Comm. Alg. 35 (2007), no. 6, 1902--1913; Link to journal, Link to arXiv

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