Normal crossing properties of complex hypersurfaces via logarithmic residues. (English) Zbl 1314.32043

Granger, Michel; Schulze, Mathias

Compos. Math. 150, No. 9, 1607-1622 (2014).

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.

Reviewer: Aleksandr G. Aleksandrov (Moskva)

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

Keywords:

normal crossing divisors; Jacobian ideal; logarithmic residue; local duality; free divisors; weakly holomorphic functions

Citations:

Zbl 0496.32007; Zbl 1333.32034

References:

[3]	doi:10.1007/BF01243911 · Zbl 0772.32023
[4]	doi:10.1017/CBO9781107325661
[7]	doi:10.1216/JCA-2013-5-1-17 · Zbl 1280.32016
[8]	doi:10.1090/S0273-0979-1995-00557-4 · Zbl 0815.52012
[9]	doi:10.1007/BFb0064400
[11]	doi:10.1016/S0764-4442(01)02166-8 · Zbl 0999.32003
[16]	doi:10.1007/BF01406236 · Zbl 0238.20034
[17]	doi:10.1007/BF02684692 · Zbl 0219.14007
[18]	doi:10.1007/BF01444547 · Zbl 0715.32013
[19]	doi:10.1007/BF02570469 · Zbl 0858.32031
[20]	doi:10.3792/pjaa.56.389 · Zbl 0476.14016
[22]	doi:10.1007/BF01362668 · Zbl 0136.20704
[23]	doi:10.1007/BF01405360 · Zbl 0224.32011
[24]	doi:10.1007/BF02406742 · JFM 19.0275.01
[26]	doi:10.1007/BF01277606 · Zbl 0608.14033
[27]	doi:10.1007/BFb0090254
[28]	doi:10.1007/BF01399504 · Zbl 0499.32008
[29]	doi:10.1017/CBO9780511662720
[30]	doi:10.1090/S0002-9939-1969-0237511-0
[32]	doi:10.1007/BF02384367 · Zbl 0553.14006
[33]	doi:10.1007/BF01163171 · Zbl 0543.14014
[34]	doi:10.1007/BFb0059377 · Zbl 0231.13009
[35]	doi:10.1007/BF01467074 · Zbl 0431.14004
[36]	doi:10.1007/978-1-4757-3849-0
[37]	doi:10.1112/plms/pdq046 · Zbl 1231.14042
[38]	doi:10.2307/1971204 · Zbl 0406.14008
[39]	doi:10.1112/S0010437X09004217 · Zbl 1238.32022