Multiplier ideals of sums via cellular resolutions

Abstract

Fix nonzero ideal sheaves $\fa_1,\dots,\fa_r$ and $\fb$ on a normal $\QQ$-Gorenstein complex variety~$X$. For any positive real numbers $\alpha$ and~$\beta$, we construct a resolution of the multiplier ideal $\JJ((\fa_1+\cdots+\fa_r)^\alpha \fb^\beta)$ by sheaves that are direct sums of multiplier ideals $\JJ(\fa_1^{\lambda_1}\cdots\fa_r^{\lambda_r}\fb^\beta)$ for various $\lambda \in \RR^r_{\geq 0}$ satisfying $\sum_{i=1}^r \lambda_i = \alpha$. 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~$\Delta$ of the simplex of nonnegative real vectors $\lambda \in \RR^r$ with $\sum_{i=1}^r \lambda_i = \alpha$. The acyclicity of our resolution reduces to that of a cellular free resolution, supported on~$\Delta$, of a related monomial ideal. Our resolution implies the multiplier ideal sum formula \[ \JJ(X,(\fa_1+\cdots+\fa_r)^\alpha \fb^\beta) = \sum_{\lambda_1+\cdots+\lambda_r=\alpha} \JJ(X,\fa_1^{\lambda_1}\cdots\fa_r^{\lambda_r}\fb^\beta), \] generalizing Takagi's formula for two summands \cite{Takagi}, and recovering Howald's multiplier ideal formula for monomial ideals~\cite{Howald} as a special case. Our resolution also yields a new exactness proof for the \emph{Skoda complex} \mbox{\cite[Section~9.6.C]{Laz}}.