Normal crossing properties of complex hypersurfaces via logarithmic residues
aa r X i v : . [ m a t h . AG ] M a y NORMAL CROSSING PROPERTIES OF COMPLEXHYPERSURFACES VIA LOGARITHMIC RESIDUES
MICHEL GRANGER AND MATHIAS SCHULZE
Abstract.
We introduce a dual logarithmic residue map for hypersurface sin-gularities and use it to answer a question of Kyoji Saito. Our result extends atheorem of Lˆe and Saito by an algebraic characterization of hypersurfaces thatare normal crossing in codimension one. For free divisors, we relate the lat-ter condition to other natural conditions involving the Jacobian ideal and thenormalization. This leads to an algebraic characterization of normal crossingdivisors. As a side result, we describe all free divisors with Gorenstein singularlocus.
Contents
1. Introduction 12. Logarithmic modules and fractional ideals 53. Logarithmic residues and duality 74. Algebraic normal crossing conditions 115. Gorenstein singular locus 13Acknowledgements 14References 151.
Introduction
Let S be a complex manifold and D be a (reduced) hypersurface D , referred to asa divisor in the sequel. In the landmark paper [Sai80], Kyoji Saito introduced thesheaves of O S -modules of logarithmic differential forms and logarithmic vector fieldson S along D . Logarithmic vector fields are tangent to D at any smooth point of D ;logarithmic differential forms have simple poles and form a complex under the usualdifferential. Saito’s clean algebraically flavored definition encodes deep geometric,topological, and representation-theoretic information on the singularities that isyet only partly understood. The precise target for his theory was the Gauß–Maninconnection on the base S of the semiuniversal deformation of isolated hypersurfacesingularities, a logarithmic connection along the discriminant D . Saito developedmainly three aspects of his logarithmic theory in [Sai80]: free divisors, logarithmicstratifications, and logarithmic residues. Mathematics Subject Classification.
Key words and phrases. logarithmic residue, duality, free divisor, normal crossing.The research leading to these results has received funding from the People Programme (MarieCurie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) underREA grant agreement n o PCIG12-GA-2012-334355.
A divisor is called free if the sheaf of logarithmic vector fields, or its dual, thesheaf of logarithmic 1-forms, is a vector bundle; in particular, normal crossing divi-sors are free. Not surprisingly, discriminants of isolated hypersurface singularitiesare free divisors (see [Sai80, (3.19)]). Similar results were shown for isolated com-plete intersection singularities (see [Loo84, § § K ( π, C (see [Poi87]). Later, the concept was generalized byde Rham and Leray to residues of closed meromorphic p -forms with simple polesalong a smooth divisor D ; these residues are holomorphic ( p − D (see[Ler59]). Residues of logarithmic differential forms along singular namely normalcrossing divisors first appeared in Deligne’s construction of the mixed Hodge struc-ture on the cohomology of smooth possibly non-compact complex varieties (see[Del71, § D , the residue of a logarithmic p -form becomes a meromorphic ( p − D , or on its normalization e D . Us-ing work of Barlet [Bar78], Aleksandrov linked Saito’s construction to Grothendieckduality theory: the image of Saito’s logarithmic residue map is the module of reg-ular differential forms on D (see [Ale88, §
4, Thm.] and [Bar78]). With Tsikh, hesuggested a generalization for complete intersection singularities based on multi-logarithmic differential forms depending on a choice of generators of the definingideal (see [AT01]). Recently, he approached the natural problem of describing themixed Hodge structure on the complement of certain divisors in terms of logarith-mic differential forms (see [Ale12, § ORMAL CROSSING PROPERTIES OF COMPLEX HYPERSURFACES 3 on e D , with holomorphic functions on e D . The latter can also be considered as weaklyholomorphic functions on D , that is, functions on the complement of the singularlocus Z of D , locally bounded near points of Z . While any such weakly holomorphicfunction is the residue of some logarithmic 1-form, the image of the residue mapcan contain functions which are not weakly holomorphic. The algebraic conditionof equality was related by Saito to a geometric and a topological property as follows(see [Sai80, (2.13)]). Theorem 1.1 (Saito) . For a divisor D in a complex manifold S , consider thefollowing conditions:(A) the local fundamental groups of the complement S \ D are Abelian;(B) in codimension one, that is, outside of an analytic subset of codimension atleast in D , D is normal crossing;(C) the residue of any logarithmic -form along D is a weakly holomorphicfunction on D .Then the implications (A) ⇒ (B) ⇒ (C) hold true. Saito asked whether the the converse implications in Theorem 1.1 hold true.The first one was later established by Lˆe and Saito [LS84]; it generalizes the Zariskiconjecture for complex plane projective nodal curves proved by Fulton and Deligne(see [Ful80, Del81]).
Theorem 1.2 (Lˆe–Saito) . The implication (A) ⇐ (B) in Theorem 1.1 holds true. Our duality of logarithmic residues turns out to translate condition (C) in The-orem 1.1 into the more familiar equality of the Jacobian ideal and the conductorideal. A result of Ragni Piene [Pie79] proves that such an equality forces D to haveonly smooth components if it has a smooth normalization. This is a technical keypoint which leads to a proof of the missing implication in Theorem 1.1. Theorem 1.3.
The implication (B) ⇐ (C) in Theorem 1.1 holds true: if theresidue of any logarithmic -form along D is a weakly holomorphic function on D then D is normal crossing in codimension one.Remark . The logarithmic stratification of S mentioned in the introduction con-sists of immersed integral manifolds of logarithmic vector fields along D (see [Sai71, § S is not locally finite, in general. Saito attached the term holonomic to this addi-tional feature: a point in S is holonomic if a neighborhood meets only finitely manylogarithmic strata. Along any logarithmic stratum, the pair ( D, S ) is analyticallytrivial which turns holonomicity into a property of logarithmic strata. The logarith-mic vector fields are tangent to the strata of the canonical Whitney stratification;the largest codimension up to which all Whitney strata are (necessarily holonomic)logarithmic strata is called the holonomic codimension (see [DM91, p. 221]).Saito [Sai80, (2.11)] proved Theorem 1.3 for plane curves. If D has holonomiccodimension at least one, this yields the general case by analytic triviality alonglogarithmic strata (see [Sai80, § xy ( x + y )( x + yz ) = 0 defines a well-known free divisor with holonomiccodimension zero. M. GRANGER AND M. SCHULZE
The preceding results and underlying techniques serve to address two naturalquestions: the algebraic characterization of condition (C) through Theorem 1.3raises the question about the algebraic characterization of normal crossing divisors.Eleonore Faber was working on this question at the same time as the results pre-sented here were developed. She considered freeness as a first approximation forbeing normal crossing and noted that normal crossing divisors satisfy an extraor-dinary condition: the ideal of partial derivatives of a defining equation is radical.She proved the following converse implications (see [Fab11, Fab12]).
Theorem 1.5 (Faber) . Consider the following conditions:(D) at any point p ∈ D , there is a local defining equation h for D such that theideal J h of partial derivatives is radical;(E) D is normal crossing.Then the following hold:(1) if D is free and satisfies condition (D) then the same holds for all irreduciblecomponents of D ;(2) conditions (D) and (E) are equivalent if D is locally a plane curve or ahyperplane arrangement, or if its singular locus is Gorenstein;(3) condition (D) implies that D is Euler homogeneous;(4) if D is free, then condition (C) implies that D is Euler homogeneous. Motivated by Faber’s problem we prove the following result.
Theorem 1.6.
Extend the list of conditions in Theorem 1.1 as follows:(F) the Jacobian ideal J D of D is radical;(G) the Jacobian ideal J D of D equals the conductor ideal C D of the normal-ization e D .Then condition (F) implies condition (B) . If D is a free divisor then conditions (B) , (F) , and (G) are equivalent.Remark . Note that J h is an O S -ideal sheaf depending on a choice of localdefining equation whereas its image J D in O D is intrinsic to D . However, by parts(3) and (4) of Theorem 1.5, condition (D) implies condition (F) and equivalenceholds if D is free.We obtain the following algebraic characterization of normal crossing divisors. Theorem 1.8.
For a free divisor with smooth normalization, any one of the con-ditions (A) , (B) , (C) , (F) , or (G) implies condition (E) .Remark . The implication (F) ⇒ (E) in Theorem 1.8 improves Theorem A in[Fab12] (see Remark 1.7), which is proved using [Pie79] like in the proof of our mainresult. Proposition C in [Fab12] is the implication (C) ⇒ (E) in Theorem 1.8, forthe proof of which Faber uses our arguments.As remarked above, free divisors are characterized by their singular loci being(empty or) maximal Cohen–Macaulay. It is natural to ask when the singular locusof a divisor is Gorenstein. This question is answered by the following theorem. Theorem 1.10.
A divisor D has Gorenstein singular locus Z of codimension if and only if D is locally the product of a quasihomogeneous plane curve and asmooth space. In particular, D is locally quasihomogeneous and Z is locally acomplete intersection. ORMAL CROSSING PROPERTIES OF COMPLEX HYPERSURFACES 5
Remark . Theorem 1.10 complements a result of Kunz–Waldi [KW84, Satz 2]saying that a Gorenstein algebroid curve has Gorenstein singular locus if and onlyif it is quasihomogeneous.2.
Logarithmic modules and fractional ideals
In this section, we review Saito’s logarithmic modules, the relation of freenessand Cohen–Macaulayness of the Jacobian ideal, and the duality of maximal Cohen–Macaulay fractional ideals. We switch to a local setup for the remainder of thearticle.Let D be a reduced effective divisor defined by I D = O S · h in the smoothcomplex analytic space germ S = ( C n , h : S → T = ( C ,
0) a functiongerm generating the ideal I D = O S · h of D . We abbreviate byΘ S := Der C ( O S ) = Hom O S (Ω S , O S )the O S -module of vector fields on S . Recall Saito’s definition [Sai80, §
1] of the O S -modules of logarithmic differential forms and of logarithmic vector fields. Definition 2.1 (Saito) . We setΩ p (log D ) := { ω ∈ Ω pS ( D ) | dω ∈ Ω p +1 S ( D ) } Der( − log D ) := { δ ∈ Θ S | dh ( δ ) ∈ I D } These modules are stalks of analogously defined coherent sheaves of O S -modules(see [Sai80, (1.3),(1.5)]). It is obvious that each of these sheaves L is torsion freeand normal, and hence reflexive (see [Har80, Prop. 1.6]). More precisely, Ω (log D )and Der( − log D ) are mutually O S -dual (see [Sai80, (1.6)]). Normality of a sheaf L means that L = i ∗ i ∗ L where i : S \ Z ֒ → S denotes the inclusion of the complementof the singular locus of D . In the case of L = Der( − log D ), this means that δ ∈ Der( − log D ) if and only if δ is tangent to D at all smooth points. In addition,Ω • (log D ) is an exterior algebra over O S closed under exterior differentiation andDer( − log D ) is closed under the Lie bracket. Definition 2.2.
A divisor D is called free if Der( − log D ) is a free O S -module.The definition of Der( − log D ) can be rephrased as a short exact sequence of O S -modules(2.1) 0 J D o o Θ Sdh o o Der( − log D ) o o o o where the Jacobian ideal J D of D is defined as the Fitting ideal J D := F n − O D (Ω D ) = (cid:28) ∂h∂x , . . . , ∂h∂x n (cid:29) ⊂ O D . Note that J D is an ideal in O D and pulls back to D h, ∂h∂x , . . . , ∂h∂x n E in O S . Weshall consider the singular locus Z of D equipped with the structure defined by J D , that is,(2.2) O Z := O D / J D . Note that Z might be non-reduced. There is the following intrinsic characterizationof free divisors in terms of their singular locus (see [Ale88, § M. GRANGER AND M. SCHULZE
Theorem 2.3.
The following are equivalent:(1) D is a free divisor;(2) J D is a maximal Cohen–Macaulay O D -module;(3) D is smooth or Z is Cohen–Macaulay of codimension one.Proof. If dh (Θ S ) does not minimally generate J D , then D ∼ = D ′ × ( C k , k >
0, bythe triviality lemma [Sai80, (3.5)]. By replacing D by D ′ , we may therefore assumethat (2.1) is a minimal resolution of J D as O S -module. Thus, the equivalence of(1) and (2) is due to the Auslander–Buchsbaum formula. By Lemma 2.6 below, J D has height at least one and the implication (2) ⇔ (3) is proved in [HK71,Satz 4.13]. (cid:3) Corollary 2.4.
Any D is free in codimension one.Proof. By Theorem 2.3, the non-free locus of D is contained in Z and equals { z ∈ Z | depth O Z,z < n − } ⊂ D. By Scheja [Sch64, Satz 5], this is an analytic set of codimension at least two in D . (cid:3) We denote by Q ( − ) the total quotient ring. Then M D := Q ( O D ) is the ring ofmeromorphic functions on D . Definition 2.5.
A fractional ideal (on D ) is a finite O D -submodule of M D whichcontains a non-zero divisor. Lemma 2.6. J D is a fractional ideal.Proof. By assumption D is reduced, so O D satisfies Serre’s condition ( R ). Thismeans that Z ⊂ D has codimension at least one. In other words, J D p for all p ∈ Ass( O D ) where the latter denotes the set of (minimal) associated primes of O D .By prime avoidance, J D S p ∈ Ass O D p . But the latter is the set of zero divisorsof O D and the claim follows. (cid:3) Proposition 2.7.
The O D -dual of any fractional ideal I is again a fractional ideal I ∨ = { f ∈ M D | f · I ⊆ O D } . The duality functor − ∨ = Hom O D ( − , O D ) reverses inclusions. It is an involution on the class of maximal Cohen–Macaulayfractional ideals.Proof. See [dJvS90, Prop. (1.7)]. (cid:3)
For lack of direct reference, we record the following consequence of the Evans–Griffith theorem.
Lemma 2.8.
Let R be a reduced Noetherian ring which satisfies Serre’s condition ( S ) and which is Gorenstein in codimension up to one. If its normalization e R isa finite R -module then it is reflexive.Proof. Let C := Hom R ( e R, R ) denote the conductor. By finiteness of ˜ R , End R ( C ) ⊆ e R and equality holds since C is an e R -ideal. By [BH93, Exc. 1.4.19], with R also C ORMAL CROSSING PROPERTIES OF COMPLEX HYPERSURFACES 7 and hence e R satisfies ( S ). Indeed, for any p ∈ Spec R ,depth e R p = depth End R p ( C p ) ≥ min { , depth C p } = min { , depth Hom R p ( e R p , R p ) }≥ min { , depth R p } . By [EG85, Thm. 3.6], this means that e R is a reflexive R -module as claimed. (cid:3) Logarithmic residues and duality
In this section, we develop the dual picture of Saito’s residue map and apply itto find inclusion relations of certain natural fractional ideals and their duals.Let π : e D → D denote the normalization of D . Then M D = M e D := Q ( O e D ) and O e D is the ring of weakly holomorphic functions on D (see [dJP00, Exc. 4.4.16.(3),Thm. 4.4.15]). Let Ω p (log D ) ρ pD / / Ω p − D ⊗ O D M D be Saito’s residue map [Sai80, §
2] which is defined as follows: by [Sai80, (1.1)], any ω ∈ Ω p (log D ) can be written as(3.1) ω = dhh ∧ ξg + ηg , for some ξ ∈ Ω p − S , η ∈ Ω pS , and g ∈ O S , which restricts to a non-zero divisor in O D . Then(3.2) ρ pD ( ω ) := ξg | D is well defined by [Sai80, (2.4)]. We shall abbreviate ρ D := ρ D and denote its imageby R D := ρ D (Ω (log D )) . Using this notation, condition (C) in Theorem 1.1, that the residue of any ω ∈ Ω (log D ) is weakly holomorphic, can be written as O e D = R D .The following result of Saito [Sai80, (2.9)] can be considered as a kind of ap-proximation of our main result Theorem 1.3; in fact, it shall be used in its proof.Combined with his freeness criterion [Sai80, (1.8)] it yields Faber’s characterizationof normal crossing divisors in Proposition B of [Fab12]. Theorem 3.1.
Let D i = { h i = 0 } , i = 1 , . . . , k , denote irreducible components of D . Then the following conditions are equivalent:(1) Ω (log D ) = P ki =1 O S dh i h i + Ω S ;(2) Ω (log D ) is generated by closed forms;(3) the D i are normal and intersect transversally in codimension one;(4) R D = L ki =1 O D i .Example . The Whitney umbrella (which is not a free divisor) satisfies the con-ditions in Theorem 1.1, but not those in Theorem 3.1.The implication (3) ⇐ (4) in Theorem 3.1 will be used in the proof of Theorem 1.3after a reduction to nearby germs with smooth irreducible components. Its proofessentially relies on parts (2) and (3) of the following Example . M. GRANGER AND M. SCHULZE (1) Let D = { xy = 0 } be a normal crossing of two irreducible components. Then dxx ∈ Ω (log D ) and ( x + y ) dxx = y d ( xy ) xy + dx − dy shows that ρ D (cid:18) dxx (cid:19) = yx + y (cid:12)(cid:12)(cid:12) D . On the components D = { x = 0 } and D = { y = 0 } of the normalization e D = D ` D , this function equals the constant function 1 and 0, respectively, and istherefore not in O D . By symmetry, this yields R D = O e D = O D × O D = J ∨ D since J D = h x, y i O D is the maximal ideal in O D = C { x, y } / h xy i . This observationwill be generalized in Proposition 3.4.(2) Conversely, assume that D = { h = x = 0 } and D = { h = x + y m = 0 } are two smooth irreducible components of D . Consider the logarithmic 1-form ω = ydx − mxdyx ( x + y m ) = y − m (cid:18) dh h − dh h (cid:19) ∈ Ω (log( D + D )) ⊂ Ω (log D ) . Its residue ρ D ( ω ) | D = y − m | D has a pole along D ∩ D unless m = 1. Thus, if O e D = R D , then D and D must intersect transversally.(3) Assume that D contains D ′ = D ∪ D ∪ D with D = { x = 0 } , D = { y =0 } , and D = { x − y = 0 } . Consider the logarithmic 1-form ω = 1 x − y (cid:18) dxx − dyy (cid:19) ∈ Ω (log D ′ ) ⊂ Ω (log D ) . Its residue ρ D ( ω ) | D = − y | D has a pole along D ∩ D ∩ D and, hence, O e D ( R D .After these preparations, we shall now approach the construction of the duallogarithmic residue. By definition, there is a short exact residue sequence(3.3) 0 / / Ω S / / Ω (log D ) ρ D / / R D / / . Applying Hom O S ( − , O S ) to (3.3) gives an exact sequence(3.4)0 Ext O S (Ω (log D ) , O S ) o o Ext O S ( R D , O S ) o o Θ S o o Der( − log D ) o o o o The right end of this sequence extends to the short exact sequence (2.1). For thehypersurface ring O D , the change of rings spectral sequence(3.5) E p,q = Ext p O D ( − , Ext q O S ( O D , O S )) ⇒ p Ext p + q O S ( − , O S )degenerates because E p,q = 0 if q = 1 and, hence,(3.6) Ext O S ( − , O S ) ∼ = E , ∼ = Hom O D ( − , O D ) ∼ = − ∨ . Therefore, the second term in the sequence (3.4) is R ∨ D . This motivates the followingkey technical result of this paper, describing the dual of Saito’s logarithmic residue. ORMAL CROSSING PROPERTIES OF COMPLEX HYPERSURFACES 9
Proposition 3.4.
There is an exact sequence (3.7)0 Ext O S (Ω (log D ) , O S ) o o R ∨ D o o Θ Sσ D o o Der( − log D ) o o o o such that σ D ( δ )( ρ D ( ω )) = dh ( δ ) · ρ D ( ω ) . In particular, σ D (Θ S ) = J D as fractionalideals. Moreover, J ∨ D = R D as fractional ideals.Proof. The spectral sequence (3.5) applied to R D is associated withRHom O S (Ω S ֒ → Ω (log D ) , h : O S → O S ) . Expanding the double complex Hom O S (Ω S ֒ → Ω (log D ) , h : O S → O S ), we obtainthe following diagram of long exact sequences. (3.8) 0 (cid:15) (cid:15) (cid:15) (cid:15) Ext O S ( R D , O S ) (cid:15) (cid:15) Hom O S (Ω S , O S ) o o h (cid:15) (cid:15) Hom O S (Ω (log D ) , O S ) o o h (cid:15) (cid:15) o o Ext O S ( R D , O S ) Hom O S (Ω S , O S ) o o (cid:15) (cid:15) Hom O S (Ω (log D ) , O S ) o o (cid:15) (cid:15) o o (cid:15) (cid:15) Hom O S (Ω S , O D ) (cid:15) (cid:15) Hom O S (Ω (log D ) , O D ) o o (cid:15) (cid:15) R ∨ Dρ ∨ D o o α (cid:15) (cid:15) o o O S (Ω (log D ) , O S ) o o Ext O S ( R D , O S ) o o (cid:15) (cid:15) We can define a homomorphism σ D from the upper left Hom O S (Ω S , O S ) to the lowerright R ∨ D by a diagram chasing process and we find that δ ∈ Θ S = Hom O S (Ω S , O S )maps to σ D ( δ ) = (cid:10) hδ, ρ − D ( − ) (cid:11) | D ∈ R ∨ D and that (3.7) is exact.By comparison with the spectral sequence, one can check that α is the changeof rings isomorphism (3.6) applied to R D , and that α ◦ σ D coincides with theconnecting homomorphism of the top row of the diagram, which is the same as theone in (3.4).Let ρ D ( ω ) ∈ R D where ω ∈ Ω (log D ). Following the definition of ρ D in (3.2),we write ω in the form (3.1). Then we compute σ D ( δ )( ρ D ( ω )) = h hδ, ω i| D = dh ( δ ) · ξg | D + h · h δ, η i g | D = dh ( δ ) · ρ D ( ω )(3.9)which proves the first two claims. For the last claim, we consider the following diagram dual to (3.8). (cid:15) (cid:15) (cid:15) (cid:15) / / Hom O S (Θ S , O S ) / / h (cid:15) (cid:15) Hom O S (Der( − log D ) , O S ) / / h (cid:15) (cid:15) Ext O S ( J D , O S ) (cid:15) (cid:15) / / Hom O S (Θ S , O S ) / / (cid:15) (cid:15) Hom O S (Der( − log D ) , O S ) / / (cid:15) (cid:15) Ext O S ( J D , O S )0 / / J ∨ Dβ (cid:15) (cid:15) dh ∨ / / Hom O S (Θ S , O D ) / / (cid:15) (cid:15) Hom O S (Der( − log D ) , O D )Ext O S ( J D , O S ) / / As before, we construct a homomorphism ρ ′ D from the upper right Hom O S (Der( − log D ) , O S )to the lower left J ∨ D such that β ◦ ρ ′ D coincides with the connecting homomorphismof the top row of the diagram, where β is the change of rings isomorphism (3.6)applied to J D . By the diagram, ω ∈ Ω (log D ) = Hom O S (Der( − log D ) , O S ) mapsto ρ ′ D ( ω ) = (cid:10) hω, dh − ( − ) (cid:11) | D ∈ J ∨ D which gives a short exact sequence(3.10) 0 / / Ω S / / Ω (log D ) ρ ′ D / / J ∨ D / / ρ ′ D ( ω )( δ ( h )) = ρ ′ D ( ω )( dh ( δ )) = h hω, δ i| D = ρ D ( ω ) · dh ( δ ) = ρ D ( ω ) · δ ( h )for any δ ( h ) ∈ J D where δ ∈ Θ S . Hence, ρ ′ D = ρ D and the last claim follows using(3.3) and (3.10). (cid:3) Corollary 3.5.
There is a chain of fractional ideals J D ⊆ R ∨ D ⊆ C D ⊆ O D ⊆ O e D ⊆ R D in M D where C D = O ∨ e D is the conductor ideal of π . In particular, J D ⊆ C D .Proof. By Lemma 2.6, J D is a fractional ideal contained in R ∨ D by Proposition 3.4.By [Sai80, (2.7),(2.8)], R D is a finite O D -module containing O e D and, hence, afractional ideal. The remaining inclusions and fractional ideals are then obtainedusing Proposition 2.7. (cid:3) Corollary 3.6. If D is free, then J D = R ∨ D as fractional ideals.Proof. If D is free, then the Ext-module in the exact sequence (3.7) disappears and σ D becomes surjective. Then the claim is part of the statement of Proposition 3.4. (cid:3) By Corollary 3.5, the inclusion O e D ⊂ R D always holds. For a free divisor D , thecase of equality is translated into more familiar terms by the following corollary. Corollary 3.7. If D is free, then R D = O e D is equivalent to J D = C D . ORMAL CROSSING PROPERTIES OF COMPLEX HYPERSURFACES 11
Proof.
By the preceding Corollary 3.6 and the last statement of Proposition 3.4,the freeness of D implies that R D and J D are mutually O D -dual. By Lemma 2.8and the definition of the conductor, the same holds true for O e D and C D . The claimfollows. (cid:3) Algebraic normal crossing conditions
In this section, we prove our main Theorem 1.3 settling the missing implicationin Theorem 1.1. We begin with some general preparations.
Lemma 4.1.
Any map φ : Y → X of analytic germs with Ω Y/X = 0 is an immer-sion.Proof.
The map φ can be embedded in a map Φ of smooth analytic germs: Y (cid:31) (cid:127) / / φ (cid:15) (cid:15) T Φ (cid:15) (cid:15) X (cid:31) (cid:127) / / S. Setting Φ i = x i ◦ Φ and φ i = Φ i + I Y for coordinates x , . . . , x n on S and I Y thedefining ideal of Y in T , we can write Φ = (Φ , . . . , Φ n ) and φ = ( φ , . . . , φ n ) andhence(4.1) Ω Y/X = Ω Y P ni =1 O Y dφ i = Ω T O T d I Y + P ni =1 O T d Φ i . We may choose T of minimal dimension so that I Y ⊆ m T and hence d I Y ⊆ m T Ω T .Now (4.1) and the hypothesis Ω Y/X = 0 show that Ω T = P ni =1 O T d Φ i + m T Ω T which implies that Ω T = P ni =1 O T d Φ i by Nakayama’s lemma. But then Φ andhence φ is a closed embedding as claimed. (cid:3) Lemma 4.2. If J D = C D and e D is smooth then D has smooth irreducible com-ponents.Proof. By definition, the ramification ideal of π is the Fitting ideal R π := F O f D (Ω e D/D ) . As a special case of a result of Ragni Piene [Pie79, Cor. 1, Prop. 1] (see also [OZ87,Cor. 2.7]), C D R π = J D O e D By hypothesis, this becomes C D R π = C D since C D is an ideal in both O D and O e D . As C D ∼ = O e D (see [MP89, Prop. 3.5.iii)]),it follows that that R π = O e D and hence that Ω e D/D = 0.Since e D is normal, irreducible and connected components coincide. By localiza-tion to a connected component e D i of e D and base change to D i = π ( e D i ) (see [Har77,Ch. II, Prop. 8.2A]), we obtain Ω e D i /D i = 0. Then the normalization e D i → D i isan immersion by Lemma 4.1 and hence D i = e D i is smooth. (cid:3) We are now ready to prove our main results.
Proof of Theorem 1.3.
In codimension one, D is free by Corollary 2.4 and hence J D = C D by Corollary 3.7 and our hypothesis. Moreover, e D is smooth in codi-mension one by normality. By our language convention, this means that there isan analytic subset A ⊂ D of codimension at least two such that, for p ∈ D \ A , J D,p = C D,p and e D is smooth above p . From Lemma 4.2 we conclude that thelocal irreducible components D i of the germ ( D, p ) are smooth. The hypothesis R D = O e D at p then reduces to the equality R D,p = L O D i . Thus, the implication(3) ⇐ (4) in Theorem 3.1 yields the claim. (cid:3) Proof of Theorem 1.6.
In order to prove that (F) implies (B), we may assume that Z is smooth and hence defined in S by two of the generators h, ∂ ( h ) , . . . , ∂ n ( h ).Then the triviality lemma [Sai80, (3.5)] shows that D is either smooth or D ∼ = C × ( C n − ,
0) and C ⊂ ( C ,
0) a plane curve. We may therefore reduce to thecase of a plane curve. Then the Mather–Yau theorem [MY82] applies (see [Fab12,Prop. 9] for details).Now assume that D is free and normal crossing in codimension one. By thefirst assumption and Theorem 2.3, Z is Cohen–Macaulay of codimension one and,in particular, satisfies Serre’s condition ( S ). By the second assumption, Z alsosatisfies Serre’s condition ( R ). Then Z is reduced, and hence J D is radical, bySerre’s reducedness criterion. This proves that (B) implies (F) for free D .The last equivalence then follows from Theorems 1.1 and 1.3 and Corollary 3.7. (cid:3) Proof of Theorem 1.8.
By Theorems 1.1, 1.3, and 1.6, we may assume that J D = C D . Then Lemma 4.2 shows that the irreducible components D i = { h i = 0 } , i = 1 , . . . , m , of D are smooth, and hence normal. It follows that R D = O e D = m M i =1 O e D i = m M i =1 O D i . By the implication (1) ⇐ (4) in Theorem 3.1, this is equivalent to(4.2) Ω (log D ) = m X i =1 O S dh i h i + Ω S . On the other hand, Saito’s criterion [Sai80, (1.8) Thm. i)] for freeness of D reads(4.3) n ^ Ω (log D ) = Ω nS ( D ) . Combining (4.2) and (4.3), it follows immediately that D is normal crossing (seealso [Fab12, Prop. B]):Considered as an O S -module and modulo Ω nS , the left-hand side of (4.3) is, dueto (4.2), generated by expressions dh i ∧ · · · ∧ dh i k ∧ dx j ∧ · · · ∧ dx j n − k h i · · · h i k , (4.4) 1 ≤ i < · · · < i k ≤ m, ≤ j < · · · < j n − k ≤ n ≥ k whereas the right-hand side of (4.3) is generated by dx ∧ · · · ∧ dx n h · · · h m . ORMAL CROSSING PROPERTIES OF COMPLEX HYPERSURFACES 13
In order for an instance of (4.4) to attain the denominator of this latter expression,it must satisfy k = m and, in particular, m ≤ n . Further, comparing numerators, dh ∧ · · · ∧ dh m ∧ dx j ∧ · · · ∧ dx j n − m must be a unit multiple of dx ∧ · · · ∧ dx n . Inother words, choosing i , . . . , i m such that { i , . . . , i m , j , . . . , j n − m } = { , . . . , n } , ∂ ( h , . . . , h m ) ∂ ( x i , . . . , x i m ) ∈ O ∗ S . By the implicit function theorem, h , . . . , h m , x j , . . . , x j n − m is then a coordinatesystem and hence D is a normal crossing divisor as claimed. (cid:3) Gorenstein singular locus
In this section, we describe the canonical module of the singular locus Z in termsof the module R D of logarithmic residues. Then, we prove Theorem 1.10.The complex of logarithmic differential forms along D relative to the map h : S → T := ( C ,
0) defining D is defined as Ω • (log h ) := Ω • (log D ) / dhh ∧ Ω •− (log D ) (see[dGMS09, §
22] or [DS12, Def. 2.7]).
Proposition 5.1.
The O Z -module R h := R D / O D fits into an exact square (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / O S h / / dh (cid:15) (cid:15) O S / / dhh (cid:15) (cid:15) O D / / (cid:15) (cid:15) / / Ω S / / (cid:15) (cid:15) Ω (log D ) ρ D / / (cid:15) (cid:15) R D / / (cid:15) (cid:15) / / Ω S/T / / (cid:15) (cid:15) Ω (log h ) / / (cid:15) (cid:15) R h / / (cid:15) (cid:15)
00 0 0 . If D is free then Z is Cohen–Macaulay with canonical module ω Z = R h ; in par-ticular, Z is Gorenstein if and only if R D is generated by and one additionalgenerator.Proof. We set ω ∅ := 0 in case D is smooth and assume Z = ∅ in the following.The exact square arises from the residue sequence (3.3) using the Snake lemma.Dualizing the short exact sequence0 / / J D / / O D / / O Z / / , one computes(5.1) Ext O D ( O Z , O D ) = J ∨ D / O D = R D / O D = R h . which shows that R h is an O Z -module. If D is free then, by Theorem 2.3, Z isCohen–Macaulay of codimension one and ω Z = R h by (5.1). (cid:3) Part (1) of the following proposition can also be found in Faber’s thesis (see[Fab11, Prop. 1.29]).
Proposition 5.2.
Let D be a free divisor. Then the following statements hold true:(1) dhh is part of an O S -basis of Ω (log D ) if and only if D is Euler homogeneous;(2) dhh is part of an O S -basis of Ω (log D ) if and only if is part of a minimalset of O D -generators of R D ;(3) R D is a cyclic O D -module if and only if D is smooth.Proof. (1) This is immediate from the existence of a dual basis and the fact that χ ∈ Der( − log D ) is an Euler vector field exactly if (cid:10) χ, dhh (cid:11) = χ ( h ) h = 1. Then χ can bechosen as a member of some basis.(2) We may assume that D ∼ = D ′ × S ′ with S ′ = ( C r ,
0) implies r = 0. Indeed,a basis of Ω (log D ) is the union of bases of Ω (log D ′ ) and Ω S ′ . This assumptionis equivalent to Der( − log D ) ⊆ m S Θ S , where m S denotes the maximal ideal of O S .Dually, this means that no basis element of Ω (log D ) can lie in Ω S .Consider a basis ω , . . . , ω n of Ω (log D ) with ω = dhh and set ρ i := ρ D ( ω i )for i = 1 , . . . , n . If ρ = 1 is not a member of some minimal set of generators of R D , then ρ = P ni =2 a i ρ i with a i ∈ m S . Thus, the form ω ′ := ω − P ni =2 a i ω i canserve as a replacement for ω in the basis. But, by construction, ρ D ( ω ′ ) = 0 whichimplies that ω ′ ∈ Ω S by (3.3). This contradicts our assumption on D .Conversely, suppose that dhh is not a member of any O S -basis ω , . . . , ω n ofΩ (log D ). Then, by Nakayama’s lemma, there are a i ∈ m S such that dhh = P ni =1 a i ω i . Applying ρ D , this gives 1 = ρ D ( dhh ) = P ni =1 a i ρ i ∈ m D R D . Againby Nakayama’s lemma, this means that 1 is not a member of any minimal set of O D -generators of R D .(3) If R D ∼ = O D , then also J D ∼ = O D by Corollary 3.6 then D must be smoothby Lipman’s criterion [Lip69]. Alternatively, it follows that J h + O S · h = O S andhence also J h = O S (see Remark 1.7); so D is smooth by the Jacobian criterion. (cid:3) As observed by Faber [Fab12, Rmk. 54], condition (2) of Proposition 5.2 issatisfied given R D = O e D (see Theorem 1.5.(4)). Proof of Theorem 1.10.
Assume that Z is Gorenstein of codimension one in D . Thepreimage J ′ D of J D in O S is then a Gorenstein ideal of height two. As such, itis a complete intersection ideal by a theorem of Serre (see [Eis95, Cor. 21.20]), andhence generated by two of the generators h, ∂ ( h ) , . . . , ∂ n ( h ). As in the proof ofTheorem 1.6, the triviality lemma [Sai80, (3.5)] shows that D is either smooth or D ∼ = C × ( C n − ,
0) and C ⊂ ( C ,
0) a plane curve. Then also C has Gorensteinsingular locus and is hence quasihomogeneous by [KW84]. Alternatively, the lastimplication follows from Propositions 5.1 and 5.2 using that quasihomogeneity of C follows from Euler homogeneity of C by Saito’s quasihomogeneity criterion [Sai71]for isolated singularities. Finally, D is quasihomogeneous and Z is a completeintersection. The converse is trivial. (cid:3) Acknowledgements
We are grateful to Eleonore Faber and David Mond for helpful discussionsand comments, and to Ragnar-Olaf Buchweitz for pointing out the statement of
ORMAL CROSSING PROPERTIES OF COMPLEX HYPERSURFACES 15
Lemma 2.8 and for outlining a proof. The foundation for this paper was laid dur-ing a “Research in Pairs” stay at the “Mathematisches Forschungsinstitut Ober-wolfach” in summer 2011. We thank the anonymous referee for careful reading andhelpful comments.
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M. Granger, Universit´e d’Angers, D´epartement de Math´ematiques, LAREMA, CNRSUMR n o E-mail address : [email protected] M. Schulze, Department of Mathematics, University of Kaiserslautern, 67663 Kaiser-slautern, Germany
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