Relative regular Riemann–Hilbert correspondence

We correct a wrong argument in Proposition 3.3. show how to get around this error. For that purpose, Sections 3.2– 3.4 of the main text have be replaced with E.3.2–E.3.4-E.3.2–E.3.4 below (Sections 3.1 and 3.5 remain unchanged). The change is new statement E.3.3 (which essentially corresponds Corollary published text) its proof. end proof Theorem 3 has been adapted correspondingly. As consequence 3, 3.3 turns out true, as well all other statements article. only modify way obtain them. E.3.3.Let M strict regular holonomic D X × S / -module X-support Z. Let Y ⊂ hypersurface containing singular locus Sing ( Z ) subsets i dim < . Then localized ∗ locally isomorphic projective pushforward relative D-module D-type. Proof.The question local. assumption on implies o : = ∖ ∩ smooth pure dimension characteristic variety | contained T By Kashiwara's equivalence, by inclusion map coherent O flat connection. strictness entails connection form ⊗ p − 1 F , d for some constant which free finite rank. One can find complex manifold ′ together divisor normal crossings morphism π → induces biholomorphism ⟶ ∼ set δ ⩽ 0 each ℓ, we consider ℓ H Although it not yet known coherent, an inductive limit (union) -submodules, hence also -coherent submodules (cf. [2, 2.1]). simply say quasi-coherent (over or over ). will use following property, deduced from similar one -modules: If ≠ sheaf-theoretic restriction zero, so owing quasi-coherence, according Since conclude now take up [18, Proof 2.11] D-type respect noticed at beginning proof, m j ↪ denote inclusion. isomorphism According 1.12, V cohomology. Furthermore, since supported ≃ After 1, improved: E.3.4. (of 1)For any rhol b complexes R Γ [ ] belong Proof.In view equivalence reduces 2.6. □ argue induction length then reduce cases projection closed embedding. first case was proved Section 1.4.1. embedding Lemma E.3.6.The P satisfies properties. Proof.It clear • Properties E.3.6(a), (b), (c), (d). Property (e) follows, adjunction, 2.9 stability S- C-constructibility under proper direct image. Last, (f) seen 3.1. End (and 1).We wish prove true ∈ proceed (E.1)). holds E.3.6(e), obviously if zero. us suppose N such k (with ⩾ let truth E.3.6(b) are reduced proving where Following notation 1.1, t (respectively f torsion part quotient) M. E.3.6(c) (applied distinguished triangle + E.3.6(f), Note ⊆ induction. Hence property assume hold. Locally (recall local E.3.6(a)), there exists satisfying assumptions E.3.3. On hand, enough check those C -c Indeed, RH belongs 2.6(b), have, [16, (3)], holonomic, apply hypothesis, true. thus strict,