Abstract
Fix nonzero ideal sheaves a 1 , . . . ., a r and b on a normal ℚ-Gorenstein complex variety X. For any positive real numbers α and β, we construct a resolution of the multiplier ideal script T((a 1 + . . . + a r ) α b β ) by sheaves that are direct sums of multiplier ideals script T(a 1 λ1 . . . a r λr b β ) for various λ ε ℝ ≥0 r satisfying Σ i=1 r λ i = α. The resolution is cellular, in the sense that its boundary maps are encoded by the algebraic chain complex of a regular CW-complex. The CW-complex is naturally expressed as a triangulation Δ of the simplex of nonnegative real vectors λ ε ℝ r with Σ i=1 r λ i = α. The acyclicity of our resolution reduces to that of a cellular free resolution, supported on Δ, of a related monomial ideal. Our resolution implies the multiplier ideal sum formula generalizing Takagi's formula for two summands [Tak05], and recovering Howald's multiplier ideal formula for monomial ideals [How01] as a special case. Our resolution also yields a new exactness proof for the Skoda complex [Laz04, Section 9.6.C]. © International Press 2008.