Introduction to Multiplier Ideals: from Analysis to Algebra

This is draft “version 0.2” of these notes. Please send comments to zteitler@ tamu. edu . To a polynomial f ∈ C[z1, . . . , zn], one can associate a family of ideals J(f, t) ⊂ C[z1, . . . , zn], for t ∈ R, t ≥ 0, called the multiplier ideals of f . The purpose of this note is to define these ideals in terms of local integrability (as in analysis) and in terms of resolution of singularities (as in algebraic geometry), and to explicate the link between these definitions. In particular we do not discuss the many applications of multiplier ideals, or the interesting properties they have. Some very well-written and more thorough treatments include [BL04], [Dem07], [Gru05], [Laz04], [Siu05]. It is my modest hope that a reader who is unfamiliar with resolution of singularities and other standard techniques of algebraic geometry might be able to go on to read one of those sources.