Multiplier ideal

In commutative algebra, the multiplier ideal associated to a sheaf of ideals over a complex variety and a real number c consists (locally) of the functions h such that

${\displaystyle {\frac {|h|^{2}}{\sum |f_{i}^{2}|^{c}}}}$

is locally integrable, where the f_i are a finite set of local generators of the ideal. Multiplier ideals were independently introduced by Nadel (1989) (who worked with sheaves over complex manifolds rather than ideals) and Lipman (1993), who called them adjoint ideals.

Multiplier ideals are discussed in the survey articles Blickle & Lazarsfeld (2004), Siu (2005), and Lazarsfeld (2009).

Algebraic geometry

In algebraic geometry, the multiplier ideal of an effective ${\displaystyle \mathbb {Q} }$-divisor measures singularities coming from the fractional parts of D. Multiplier ideals are often applied in tandem with vanishing theorems such as the Kodaira vanishing theorem and the Kawamata–Viehweg vanishing theorem.

Let X be a smooth complex variety and D an effective ${\displaystyle \mathbb {Q} }$-divisor on it. Let ${\displaystyle \mu :X'\to X}$ be a log resolution of D (e.g., Hironaka's resolution). The multiplier ideal of D is

${\displaystyle J(D)=\mu _{*}{\mathcal {O}}(K_{X'/X}-[\mu ^{*}D])}$

where ${\displaystyle K_{X'/X}}$ is the relative canonical divisor: ${\displaystyle K_{X'/X}=K_{X'}-\mu ^{*}K_{X}}$. It is an ideal sheaf of ${\displaystyle {\mathcal {O}}_{X}}$. If D is integral, then ${\displaystyle J(D)={\mathcal {O}}_{X}(-D)}$.

See also

• Canonical singularity
• Test ideal

References

• Blickle, Manuel; Lazarsfeld, Robert (2004), "An informal introduction to multiplier ideals", Trends in commutative algebra, Math. Sci. Res. Inst. Publ., 51, Cambridge University Press, pp. 87–114, CiteSeerX 10.1.1.241.4916, doi:10.1017/CBO9780511756382.004, MR 2132649
• Lazarsfeld, Robert (2009), "A short course on multiplier ideals", 2008 PCMI lectures, arXiv:0901.0651, Bibcode:2009arXiv0901.0651L
• Lazarsfeld, Robert (2004). Positivity in algebraic geometry II. Berlin: Springer-Verlag.
• Lipman, Joseph (1993), "Adjoints and polars of simple complete ideals in two-dimensional regular local rings" (PDF), Bulletin de la Société Mathématique de Belgique. Série A, 45 (1): 223–244, MR 1316244
• Nadel, Alan Michael (1989), "Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature", Proceedings of the National Academy of Sciences of the United States of America, 86 (19): 7299–7300, Bibcode:1989PNAS...86.7299N, doi:10.1073/pnas.86.19.7299, JSTOR 34630, MR 1015491, PMC 298048, PMID 16594070
• Siu, Yum-Tong (2005), "Multiplier ideal sheaves in complex and algebraic geometry", Science China Mathematics, 48: 1–31, arXiv:math/0504259, Bibcode:2005ScChA..48....1S, doi:10.1007/BF02884693, MR 2156488