The multiplier ideals of a sum of ideals

Abstract

We prove that if a _ \underline {\mathbf {a}} , b _ ⊆ O X \underline {\mathbf {b}}\subseteq \mathcal {O}_X are nonzero sheaves of ideals on a complex smooth variety X X , then for every γ ∈ Q + \gamma \in {\mathbb Q}_+ we have the following relation between the multiplier ideals of a _ \underline {\mathbf {a}} , b _ \underline {\mathbf {b}} and a _ + b _ \underline {\mathbf {a}}+\underline {\mathbf {b}} : I ( X , γ ⋅ ( a _ + b _ ) ) ⊆ ∑ α + β = γ I ( X , α ⋅ a _ ) ⋅ I ( X , β ⋅ 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.