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 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 is locally integrable, where the fi 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). In algebraic geometry, the multiplier ideal of an effective Q {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 Q {displaystyle mathbb {Q} } -divisor on it. Let μ : X ′ → X {displaystyle mu :X' o X} be a log resolution of D (e.g., Hironaka's resolution). The multiplier ideal of D is where K X ′ / X {displaystyle K_{X'/X}} is the relative canonical divisor: K X ′ / X = K X ′ − μ ∗ K X {displaystyle K_{X'/X}=K_{X'}-mu ^{*}K_{X}} . It is an ideal sheaf of O X {displaystyle {mathcal {O}}_{X}} . If D is integral, then J ( D ) = O X ( − D ) {displaystyle J(D)={mathcal {O}}_{X}(-D)} .