The multiplier ideals of a sum of ideals

Abstract:

We prove that if $\underline {\mathbf {a}}$, $\underline {\mathbf {b}}\subseteq \mathcal {O}_X$ are nonzero sheaves of ideals on a complex smooth variety $X$, then for every $\gamma \in {\mathbb Q}_+$ we have the following relation between the multiplier ideals of $\underline {\mathbf {a}}$, $\underline {\mathbf {b}}$ and $\underline {\mathbf {a}}+\underline {\mathbf {b}}$: \begin{equation*}\mathcal {I}\left (X,\gamma \cdot (\underline {\mathbf {a}}+ \underline {\mathbf {b}})\right )\subseteq \sum _{\alpha +\beta =\gamma } \mathcal {I}(X,\alpha \cdot \underline {\mathbf {a}})\cdot \mathcal {I}(X,\beta \cdot \underline {\mathbf {b}}).\end{equation*} A similar formula holds for the asymptotic multiplier ideals of the sum of two graded systems of ideals.

We use this result to approximate at a given point arbitrary multiplier ideals by multiplier ideals associated to zero dimensional ideals. This is applied to compare the multiplier ideals associated to a scheme in different embeddings.