Algorithms for computing multiplier ideals
Let \(X\) be a smooth complex variety and \(\mathfrak{a}\subseteq \mathcal{O}_X\) an ideal sheaf. Let \(\pi:\widetilde{X}\to X\) be a log resolution of \(\mathfrak{a}\). Then the multiplier ideal of \(\mathfrak{a}\) with exponent \(c\in \mathbb{R}_{\geq 0}\) is defined as \(J(\mathfrak{a}^c)=J(c\cdot \mathfrak{a})=\pi_\ast \mathcal{O}_{\widetilde{X}}(K_{\widetilde{X}|X}-\lfloor cF\rfloor)\subseteq \mathcal{O}_X\). Here \(K_{\widetilde{X}|X}\) is the relative canonical divisor of \(\pi\) and \(\mathfrak{a}\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-F)\) for an effective divisor \(F\) on \(\widetilde{X}\) such that Supp \(F\cup\text{Exc}(\pi)\) is a divisor with simple normal crossing. \(\text{Exc} (\pi)\) is the exceptional locus of \(\pi\). Algorithms are given for computing multiplier ideals using the theory of \(D\)-modules, i.e. Gröbner bases in Weyl algebras are used. Several examples are given.
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