Multiplier ideals of sums via cellular resolutions

Fix nonzero ideal sheaves a1, . . . ., ar 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((a1 + . . . + ar)αbβ) by sheaves that are direct sums of multiplier ideals script T(a1λ1 . . . arλrbβ) for various λ ε ℝ≥0r satisfying Σi=1r λi = 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 ℝr with Σi=1r λi = 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.

Duke Scholars

Author Ezra Miller Mathematics