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ANNALESDEL'INSTITUTFOURIERANNALES

DE

L"INSTITUT FOURIER

Delphine POL

On the values of logarithmic residues along curves

Tome 68, n

o2 (2018), p. 725-766.

© Association des Annales de l"institut Fourier, 2018,

Certains droits réservés.Cet article est mis à disposition selon les termes de la licence CREATIVECOMMONS ATTRIBUTION-PAS DE MODIFICATION3.0 FRANCE. L"accès aux articles de la revue " Annales de l"institut Fourier » (http://aif.cedram.org/), implique l"accord avec les conditions générales d"utilisation (http://aif.cedram.org/legal/). cedramArticle mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/

Ann. Inst. Fourier, Grenoble

68, 2 (2018) 725-766

ON THE VALUES OF LOGARITHMIC RESIDUES

ALONG CURVES

by Delphine POLAbstract. -We consider the germ of a reduced curve, possibly reducible. F. Delgado de la Mata proved that such a curve is Gorenstein if and only if its semigroup of values is symmetrical. We extend here this symmetry property to any fractional ideal of a Gorenstein curve. We then focus on the set of values of the module of logarithmic residues along plane curves or complete intersection curves, which determines and is determined by the values of the Jacobian ideal thanks to which are used in the analytic classification of plane branches. We also study the behaviour of logarithmic residues in an equisingular deformation of a plane curve. Résumé. -On considère un germe de courbe réduit, éventuellement réduc- tible. F. Delgado de la Mata a montré qu"une telle courbe est Gorenstein si et seulement si son semigroupe des multi-valuations est symétrique. Nous étendons ici cette propriété de symétrie à tout idéal fractionnaire d"une courbe Gorenstein. Nous nous intéressons ensuite à l"ensemble des multi-valuations du module des ré- sidus logarithmiques d"une courbe plane ou intersection complète, qui détermine et

est déterminé par les multi-valuations de l"idéal jacobien grâce à notre théorème de

sont utilisées dans la classification analytique des branches planes. Nous étudions aussi le comportement des résidus logarithmiques dans une déformation équisingu- lière de courbe plane.

1. Introduction

LetDbe the germ of a reduced hypersurface in(Cn,0)defined byf? C{x}:=C{x1,...,xn}and with ringOD=C{x}/(f). In his fundamental paper [ 28
], K. Saito introduces the notions of logarithmic vector fields, logarithmic differential forms and their residues. A logarithmic differential

form is a meromorphic form on a neighbourhood of the origin inCnwhichKeywords:logarithmic residues, duality, Gorenstein curves, values, equisingular

deformations.

2010Mathematics Subject Classification:14H20, 14B07, 32A27.

726Delphine POL

has simple poles alongDand such that its differential also has simple poles alongD. A logarithmicq-formωsatisfies: gω=dff whereg?C{x}does not induce a zero divisor inOD,ξis a holomorphic (q-1)-form andηis a holomorphicq-form. Then, the logarithmic residue res q(ω)ofωis defined as the coefficient ofdff , that is to say: res q(ω) =?ξg ?D ?Ωq-1

D?ODQ(OD),

withΩq-1 ring of fractions ofOD. We denote byRDtheOD-module of logarithmic residues of logarithmic1-forms. In [ 12 ], M. Granger and M. Schulze prove that theOD-dual of the Jaco- bian ideal ofDisRD. If in additionDis free, that is to say, if the mod- ule of logarithmic differential1-forms is a freeC{x}-module, the converse also holds: the dual ofRDis the Jacobian ideal. They use this duality to prove a characterization of normal crossing divisors in terms of logarithmic residues: if the moduleRDis equal to the module of weakly holomorphic functions onDthenDis normal crossing in codimension1. The converse implication was already proved in [ 28
The purpose of this paper is to investigate more deeply the module of logarithmic residues. We focus on the case of plane curves or complete intersection curves. Plane curves are always free divisors, and they are the only singular free divisors with isolated singularities. The notion of multi- residues along complete intersections was introduced by A. G. Aleksandrov and A. Tsikh in [ 1 In order to describe the module of residues, or more generally, any frac- tional ideal, we will consider the set of values, which is defined as follows. LetD=D1?···?Dpbe the germ of a reduced complex analytic curve with pirreducible components. The normalization of the local ringODinduces a mapval :Q(OD)→(Z? {∞})pcalled the value map, which associates with a fractiong?Q(OD)thep-uple of the valuations ofgalong each irreducible component ofD. Given a fractional idealI?Q(OD)(see Def- inition 2.2 ), we denote byval(I)the set of values of the non zero divisors ofI, andI?:={g?Q(OD);g·I?OD}the dual ofI.

Let us explain the content of Section

2 . We prove that the values of a fractional ideal and the values of its dual determine each other, and we give explicitly the relation between them. We then apply this result to the

ANNALES DE L"INSTITUT FOURIER

LOGARITHMIC RESIDUES ALONG CURVES727

case ofRDand of the Jacobian ideal ofD, denoted byJD, in Sections3 and 4 . This relation is in fact a generalization of the following well-known theorem of Kunz in the case of irreducible curves: Theorem 1.1([19]). -IfDis irreducible,ODis Gorenstein if and only if the following property, which is called a symmetry property, is satisfied: for allv?Z, v?val(OD)??γ-v-1/?val(OD), whereγis the conductor ofD, that is to say,

γ= min{α?N;α+N?val(OD)}.

In [ 21
, Theorem 2.8], F. Delgado de la Mata generalizes the former result to the case of reducible curves. He proves that a curve is Gorenstein if and only ifval(OD)satisfies a symmetry property described below. We prove here that Delgado"s symmetry has an analogue which links the values of a fractional ideal and the values of its dual. Whereas the sym- metry is immediate for irreducible curves, the proof of this generalization of Delgado"s theorem is much more subtle. It leads to the main result of this section, namely Theorem 1.2 , which generalizes Theorem 2.4 of [ 25
] to any Gorenstein curve and any fractional ideal. To give the statement of our symmetry theorem, we introduce the following notation fori? {1,...,p}, v?ZpandIa fractional ideal (see Notation2.13 ): i(v,val(I)) :={α?val(I);αi=vi,?j?=i,αj> vj}, andΔ(v,val(I)) =?p i=1Δi(v,val(I)). We consider the product order on Z p, that is to say, forα,β?Zp,α6βmeans that for alli? {1,...,p},αi6 i. The conductor ofDis:γ:= inf{α?Np;α+Np?val(OD)}. We set

1= (1,...,1). The statement of our main theorem is:

Theorem 1.2. -LetDbe the germ of a reduced analytic curve with pirreducible components. Then, the ringODof the curve is a Gorenstein ring if and only if for all fractional idealsI?Q(OD)the following property is satisfied for allv?Zp: (1.1)v?val(I?)??Δ(γ-v-1,val(I)) =∅. Delgado"s theorem concerns the caseI=OD(see Theorem2.14 ). A similar symmetry was recently proved in [ 18 ] using combinatorial methods which involve canonical modules.

In Sections

3 and 4 , we use Theorem 1.2 to study the mo duleof logarith- mic residues along complete intersection curves, with a particular attention to the case of plane curves.

TOME 68 (2018), FASCICULE 2

728Delphine POL

In Subsection

3.2 , we give some properties of the set of values of the module of logarithmic residues and of the Jacobian ideal for plane curves. We investigate the zero divisors ofRDandJD, which are described in

Propositions

3.15 and 3.18 . Thanks to our symmetry theorem, we then de- termine the conductor ofRD, which is-(m(1),...,m(p))+1, wherem(i)is the multiplicity of the branchDi. We also mention the relation between log- to this relation, we recover the result of O. Zariski on the dimension of the

In Subsection

3.3 , we recall the theory of multi-logarithmic differential forms and multi-residues along a reduced complete intersection developed by A. G. Aleksandrov and A. Tsikh in [ 1 ]. Since our symmetry theorem is true for any Gorenstein curve, it is in particular true for complete intersec- tion curves. As in the hypersurface case, we again have a duality between multi-residues and the Jacobian ideal, so that their sets of values deter- mine each other. Moreover, we prove here the following proposition (see

Proposition

3.31 for a more precise statemen t): a reduced complete intersection curveC. The values ofJCand the values ofΩ1Csatisfy: val(JC) =γ+ val(Ω1C)-1. 15 and [ 16 ] to study the problem of the analytic classification of plane curves with one or two branches. The last section is devoted to the study of the behaviour of logarithmic residues in an equisingular deformation of a plane curve. In particular, we define a stratification by the values of the logarithmic residues, which is the 3.3 We prove that this stratification is finer than the stratification by the Tju- rina number. We give an example in which the stratification by logarithmic residues is strictly finer than the stratification by the Tjurina number (see

Example

4.16 ). We show that the stratification by logarithmic residues is finite and constructible (see Propositions 4.14 and 4.15quotesdbs_dbs44.pdfusesText_44

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