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
is locally integrable, where the "f""i" are a finite set of local generators of the ideal. Multiplier ideals were independently introduced by (who worked with sheaves over complex manifolds rather than ideals) and , who called them adjoint ideals.
Multiplier ideals are discussed in the survey articles , , and .
Algebraic geometry.
In algebraic geometry, the multiplier ideal of an effective formula_2-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 formula_2-divisor on it. Let formula_4 be a log resolution of "D" (e.g., Hironaka's resolution). The multiplier ideal of "D" is
where formula_6 is the relative canonical divisor: formula_7. It is an ideal sheaf of formula_8. If "D" is integral, then formula_9.