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Definition 6.8.5990C506B7D25

One local-integrability construction with source-specific scopes: weights and componentwise minus-infinity values in Foundations, plurisubharmonic functions in SGPT, and quasi-plurisubharmonic functions in the Hermitian text. Each scope is retained below.

Definition 6.8.5.990C506B7D25 Let $\varphi $ be a weight on $X$. Its multiplier ideal is the subsheaf $\mathcal I(\varphi )\subset \mathcal O_ X$ with stalks

\[ \mathcal I(\varphi )_ x =\{ f\in \mathcal O_{X,x}: |f|^2\mathrm e^{-\varphi } \text{ is integrable near }x\} . \]

Here $|f|^2\mathrm e^{-\varphi }$ is taken to be $0$ where $f=0$, so the zero germ always belongs to $\mathcal I(\varphi )_ x$. The same definition applies to a function $\varphi $ that is plurisubharmonic or identically $-\infty $ on each component, as happens for restrictions of weights; where $\varphi $ is identically $-\infty $, the multiplier ideal is zero.

Read the alternative statement and proof

Definition 6.8.5.990C506B7D25 Let $\varphi \in \mathrm{PSH}(X)$, the multiplier ideal sheaf $\mathcal{I}(\varphi )$ of $\varphi $ is by definition the ideal sheaf given by

\[ \Gamma \left(U,\mathcal{I}(\varphi )\right)=\left\{ f\in \mathcal{O}_ X(U) : |f|^2\exp (-\varphi )\in L^1_{\mathrm{loc}}(U) \right\} \]

for any open subset $U\subseteq X$.

Read the alternative statement and proof

Definition 6.8.5.990C506B7D25 Let $\mathcal O_ X$ denote the sheaf of holomorphic functions on $X$. For a quasi-plurisubharmonic function $u$ on $X$, the multiplier ideal $\mathcal I(u)\subseteq \mathcal O_ X$ is the ideal sheaf whose sections over an open set $U$ are the holomorphic functions $g$ on $U$ for which $|g|^2\mathrm{e}^{-u}$ is locally integrable on $U$ with respect to a smooth positive volume form. Its stalk at $x$ is denoted by $\mathcal I(u)_ x$.

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