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Multiplier ideals, \(v\)-filtrations and transversal sections. (English) Zbl 1107.14003
Let \(X\) be a complex \(n\)-dimensional manifold and \(D\subset X\) an effective divisor defined by a holomorphic function \(f\). Let \(i_T:T\to X\) be the inclusion of a closed submanifold which is transversal to \(D\). Let \(M\) be a regular holonomic \(D_X\)-module such that \(T\) is non-characteristic for \(M\) and for \(M(\ast D)\). Let \(V\) denote the Kashiwara–Malgrange filtration of \(M\) along \(D\) and also the Kashiwara–Malgrange filtration of the restriction \(i^\ast_T M\) along \(T\cap D\). It is proved that for any \(\alpha\in \mathbb C\)
(1) \(i^\ast_T(V^\alpha M)= V^\alpha i^\ast_T M\)
(2) \(i^\ast_T(\text{Gr}^\alpha_V M)=\text{Gr}^\alpha_Vi^\ast_T M\).
Using the relation between multiplier ideals and \(V\)-filtration one obtains that the restriction to a smooth transversal section commutes with the computation of multiplier ideals and \(V\)-filtrations. As an application the constancy of the jumping numbers and the spectrum along any stratum of a Whitney regular stratification is proved.

MSC:
14B05 Singularities in algebraic geometry
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
32S30 Deformations of complex singularities; vanishing cycles
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
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