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Normal crossing properties of complex hypersurfaces via logarithmic residues. (English) Zbl 1314.32043
Let \(D\) be a reduced divisor in a complex manifold \(S\). In 1980 K. Saito has noted that if \(D\) is normal crossing outside an analytic subset of codimension at least two in \(D\), then the residue of any logarithmic differential 1-form along \(D\) is holomorphic on the normalization \({\widetilde{D}}\), that is, \(\text{res\,} (\Omega^1_S(\log D)) = {\mathfrak n}_*({\mathcal O}_{\widetilde{D}})\), where \({\mathfrak n} : \widetilde{D} \rightarrow D\) is the morphism of normalization (see [K. Saito, J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 265–291 (1980; Zbl 0496.32007)], Lemma \(2.13\)). There he has also stated (Theorem \(2.11\)) that the converse is true for plane curves. The paper under review contains a proof of the corresponding statement for divisors of arbitrary dimension. For this, the authors introduce a dual logarithmic residue map and then use nice relations between properties of the Jacobian ideal and the normalization studied also in detail by E. Faber [Math. Ann. 361, No. 3–4, 995–1020 (2015; Zbl 1333.32034)]. In conclusion, some interesting applications for free divisors are discussed.

MSC:
32S25 Complex surface and hypersurface singularities
32A27 Residues for several complex variables
32S10 Invariants of analytic local rings
32C37 Duality theorems for analytic spaces
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