On the local zeta functions and the b-functions of certain hyperplane arrangements
aa r X i v : . [ m a t h . AG ] M a y On the local zeta functions and the b -functionsof certain hyperplane arrangements (with Appendix by Willem Veys) Nero Budur, Morihiko Saito, and Sergey Yuzvinsky
Abstract.
Conjectures of J. Igusa for p-adic local zeta functions and of J. Denef andF. Loeser for topological local zeta functions assert that (the real part of) the polesof these local zeta functions are roots of the Bernstein-Sato polynomials (i.e. the b-functions). We prove these conjectures for certain hyperplane arrangements, includingthe case of reduced hyperplane arrangements in three-dimensional affine space.
Introduction
Let K be a p -adic field, i.e. a finite extension of Q p , and O K be the ring of integers of K .We have the norm defined by | x | K = q − v ( x ) for x ∈ K ∗ where v ( x ) ∈ Z is the valuation(or the order) of x ∈ K and q is the cardinality of the residue field O K / m K with m K the maximal ideal. For a nonconstant polynomial f ∈ K [ x , . . . , x n ], Igusa’s p -adic localzeta function (associated with the characteristic function of O nK ⊂ K n , see [Ig1], [Ig4]) isdefined by the meromorphic continuation of the integral Z pf ( s ) = Z O nK | f ( x ) | sK dx for Re s > . Here dx denotes the Haar measure on the compact open subgroup O nK of K n , which isthe p -adic analogue of the polydisk ∆ n in C n . Note that Z pf ( s ) is closely related to thePoincar´e series associated with the numbers of solutions of f = 0 in ( O K / m iK ) n for i > f ∈ O K [ x , . . . , x n ].On the other hand, the Bernstein-Sato polynomial (i.e. the b -function) of a polynomial f ∈ K [ x , . . . , x n ] is the monic polynomial b f ( s ) of the least degree satisfying the relation b f ( s ) f s = P f s +1 in R f [ s ] f s for some P ∈ D n [ s ] , where R f is the localization of R := K [ x , . . . , x n ] by f and D n is the Weyl algebra whichis generated over K by x , . . . , x n and ∂/∂x , . . . , ∂/∂x n . Here K may be any field of b f ( s ) is invariant by extensions of the field K , see (2.1) below. (Thereis a shift of the variable s by 1 if one uses the definition of the Bernstein polynomial in [Be]since f s is replaced by f s − there). The local b -function b f,x ( s ) is defined by replacingthe Weyl algebra D n with D X,x . Note that for a homogeneous polynomial f , we have b f ( s ) = b f, ( s ).A conjecture of J. Igusa [Ig2] asserts the following. Conjecture (A) p . The real part of any pole of the p -adic local zeta function Z pf ( s ) is aroot of b f ( s ).Inspired by this conjecture, J. Denef and F. Loeser [DL] defined the topological localzeta function Z topf,x ( s ) (see (1.1.1) below) for a nonconstant polynomial f and x ∈ f − (0)in the case K = C , and conjectured the following. Conjecture (A) top . Any pole of the topological local zeta function Z topf,x ( s ) is a root of b f,x ( s ).There is a weaker version of the conjectures, due to Igusa, and Denef and Loeserrespectively, and called the monodromy conjecture , as follows. Conjecture (B) p . For any pole α of the p -adic local zeta function Z f ( s ), e πi Re( α ) isan eigenvalue of the Milnor monodromy of f C at some x ∈ f − C (0) ⊂ C n choosing anembedding K ֒ → C , where f C is the image of f in C [ x , . . . , x n ]. Conjecture (B) top . For any pole α of the topological local zeta function Z topf,x ( s ), e πiα is an eigenvalue of the Milnor monodromy of f at y ∈ f − (0) sufficiently near x .In Conjecture (B) p , it is enough to consider an embedding K f ֒ → C , where K f is thesubfield of K generated by the coefficients of the linear factors of f so that D is defined over K f . Originally Conjecture (A) p and (B) p are stated for a polynomial f ∈ F [ x , . . . , x n ]with F a number field and K the completion of F at a prime of F (except possibly for afinite number of primes). In the hyperplane arrangement case, however, this assumptiondoes not seem to be essential since Conjecture (B) p is already proved by [BMT] andConjecture (A) p is reduced to Conjecture (C) below.By Conjecture (A) we will mean Conjecture (A) p or Conjecture (A) top depending onwhether K is the p -adic or complex number field, and similarly for Conjecture (B). Notethat the eigenvalues of the Milnor monodromies in Conjecture (B) can be defined in apurely algebraic way using the V -filtration of Kashiwara [Ka2] and Malgrange [Ma2] onthe D n [ s ]-module R f [ s ] f s , and the union of the eigenvalues of the Milnor monodromiesfor x ∈ f − C (0) ⊂ C n is independent of the choice of the embedding K f ֒ → C , see (2.1)below. Moreover we have the following. Proposition 1.
Let K be a subfield of C , and f ∈ K [ x , . . . , x n ] . ( i ) A complex number λ ∈ C is an eigenvalue of the Milnor monodromy of f C at some x ∈ f − C (0) ⊂ C n if and only if there is a root α of b f ( s ) such that λ = e − πiα . ( ii ) If K = C , then for any x ∈ f − (0) , there is an open neighborhood U of x in classicaltopology such that for any open neighborhood U ′ of x in U , the following two conditionsare equivalent. a ) The number λ is an eigenvalue of the Milnor monodromy of f at some y ∈ f − (0) ∩ U ′ . ( b ) There is a root α of b f,x ( s ) such that λ = e − πiα . This follows from [Ka1], [Ma2]. By Proposition 1, Conjecture (B) can be viewed asthe modulo Z version of Conjecture (A), and is weaker than the latter. It is known thatConjectures (A) and (B) are rather difficult to prove, see e.g. [ACLM1], [ACLM2], [Den],[DL], [Ig3], [Ig4], [KSZ], [Lo1], [Lo2], [LVa], [LV1], [LV2], [Ro], [VP], [Ve1], [Ve2], [Ve3],[Ve4]. For a generalization to the ideal case, see [HMY], [VV] (using [BMS]).In this paper we prove Conjecture (A) for certain affine hyperplane arrangements D in K n . Let D i be the irreducible components of D , and m i be the multiplicity of D along D i . Let f be a defining equation of D . Set d := deg D = deg f = P i m i . In [BMT],Conjecture (A) is reduced to the following. Conjecture (C) . Let D be an indecomposable essential central hyperplane arrangementin C n with degree d . Then b f ( − n/d ) = 0.Here central and essential respectively mean that 0 ∈ D i for any i and dim T i D i = 0.We say that D is indecomposable if it is not a union of the pullbacks of arrangements bythe two projections of some decomposition C n = C n ′ × C n ′′ as a vector space. Note thatthe proof of Conjecture (B) in [BMT] implies that − n/d − b f ( s ) in case − n/d is not, since the roots of b f ( s ) are in ( − , Theorem 1 [BMT].
For an affine hyperplane arrangement D in K n , Conjecture ( A ) holdsif Conjecture ( C ) for ( D/L ) C holds for every dense edge L of D . Here an edge means an intersection of D i , and D/L denotes the arrangement in K n /L defined by the D i containing L and with the same multiplicity m i , where we may assume0 ∈ L replacing the origin of K n if necessary. We call an edge L = K n dense if D/L isindecomposable. If K is a p -adic filed, then ( D/L ) C denotes the scalar extension of D/L defined by choosing an embedding K f ֒ → C where K f ⊂ K is the smallest subfield suchthat f and all the D i are defined over K f . We have ( D/L ) C = D/L in the case K = C .Theorem 1 is proved by using a resolution of singularities obtained by blowing uponly the proper transforms of the dense edges in [STV] (together with Igusa’s calculationof candidates for poles of the p -adic zeta functions [Ig1] in the p -adic case, see also (1.1.3)below). Because of this very special kind of resolution, all the obtained candidates forpoles contribute at least to the monodromy eigenvalues, and Conjecture (B) is proved in[BMT] for all the candidates for poles using the calculation of the Milnor cohomology ofhyperplane arrangements in [CS] (or [Di], Prop. 6.4.6) together with a result of [STV] onthe relation between indecomposability and nonvanishing of the Euler characteristic of theprojective complement. This is contrary to the most other cases where lots of cancelationsof apparent poles occur, see [Den], [Lo1], [Ve1], [Ve2], [Ve3] (and Remark (1.2) below).Recently W. Veys informed us that there are examples of hyperplane arrangement ofdegree d in C n such that − n/d is not a pole of Z topf, ( s ) in the case n = 3 with D non-reduced or n = 5 with D reduced, see Appendix. These examples imply a negative answerto Question (Q) in (1.4). There are no such examples if n = 2 or n = 3 with D reduced,see Propositions (1.5) and (1.8) below. 3n this paper we prove the following. Theorem 2.
Conjecture ( C ) holds in the following cases. ( i ) { } is a good dense edge of D . ( ii ) D is reduced with n ≤ . ( iii ) D is reduced, ( n, d ) = 1 , and D d is generic relative to the other D j ( j = d ) . Here L is called a good dense edge if for any dense edges L ′ ⊃ L , we have n ( L ) /d ( L ) ≤ n ( L ′ ) /d ( L ′ ) , where d ( L ) = mult L D = P D i ⊃ L m i and n ( L ) = codim L . We say that D d is genericrelative to the other D j ( j = d ) if any nonzero intersection of D j ( j = d ) is not containedin D d , see [FT], Example 4.5.In case (i), Theorem 2 follows from Teitler’s refinement [Te] of Mustat¸ˇa’s formula [Mu]for multiplier ideals using only dense edges, together with a well-known relation betweenthe jumping coefficients and the roots of b f ( s ), see [ELSV]. In case (ii) or (iii), we use ageneralization of Malgrange’s formula for the roots of b f ( s ) in the isolated singularity case(see [Sa1], [Sa2]) reducing the assertion to a certain combinatorial problem which can besolved under condition (ii) or (iii), where we need a result from [FT] in case (iii).Combining Theorems 1 and 2, we get Theorem 3.
For an affine hyperplane arrangement D in K n , conjecture ( A ) holds if forevery dense edge L of D , one of the three conditions in Theorem is satisfied for ( D/L ) C .In particular, Conjecture ( A ) holds in the following cases. ( i ) D is of moderate type. ( ii ) D is reduced with n ≤ . ( iii ) D is reduced with n = 4 , and for each -dimensional dense edge L of D , eithercondition ( ii ) or ( iii ) in Theorem is satisfied for ( D/L ) C . Here D is called of moderate type if all the dense edges are good. Note that in thecase (iii), condition (ii) in Theorem 2 is satisfied for ( D/L ) C with L = 0. It seems quitedifficult to generalize the arguments in this paper to the non-reduced case even for n = 3,or to the 4-dimensional case even for reduced D .We would like to thank W. Veys for useful comments and especially for examples inAppendix solving Question (Q) in (1.4) negatively.In Section 1 we recall some facts from the theory of local zeta functions. In Section2 we explain how to calculate the b -functions of homogeneous polynomials, and proveTheorem 2 in cases (i) and (iii). In Section 2 we prove Theorem 2 in case (ii). In Appendixby W. Veys, we describe some examples related to Question (Q) in (1.4).4 . Local zeta functions1.1. Let K be the complex or p -adic number field. Let X be a complex manifold ofdimension n with f a holomorphic function on X if K = C , and X = K n with f ∈ K [ x , . . . , x n ] if K is a p -adic field. Set D = f − (0). Let σ : ( e X, E ) → ( X, D ) be anembedded resolution with E j the irreducible components of E := σ ∗ D . Set E ◦ I = T i ∈ I E j \ S i / ∈ I E j , m j = mult E j σ ∗ D, r j = mult E j det(Jac( σ )) . If K = C , the topological local zeta function for x ∈ D is defined by(1 . . Z topf,x ( s ) = X I χ ( E ◦ I ∩ σ − ( x )) Y j ∈ I m j s + r j + 1 , which is independent of the choice of the resolution (see [DL]). So we get candidates forpoles(1 . . α j := − r j + 1 m j . Note that each α j is not necessarily a pole of Z topf,x ( s ) in general. It is not easy to determineexactly false poles since there are cancelations of poles in many cases, see [Den], [Lo1], [Ve1],[Ve2], [Ve3] (and Remark (1.2) below). In the hyperplane arrangement case, however, thereis a special kind of resolution by [STV] so that Conjecture (B) is proved for the abovecandidates for poles although it is still unclear whether they are really poles.The situation is similar in the p -adic case where Igusa’s calculation (see e.g. [Ig4],Theorem 8.2.1 or [Den]) implies that the poles of the local zeta function are among thecomplex numbers(1 . . α j,k := − r j + 1 m j − π √− km j log q ( k ∈ Z ) . It is known that there are remarkable cancelations of poles by the summa-tion in the definition (1.1.1). So it is not easy to eliminate false poles, although the curvecase is rather well understood, see [Den], [Lo1], [Ve1], [Ve2], [Ve3]. (For a relatively simpleproof of Conjecture (B) for n = 2, see [Ro].) It is also known that only a few of the rootsof b f ( s ) can be detected by the local zeta function. Let D be a hyperplane arrangement defined by a polynomial f . Thenthe topological local zeta function Z topf,x ( s ) is a combinatorial invariant.Proof. By the definition of Z topf,x ( s ) in (1.1.1), we may assume D is central, x = 0. Wehave to calculate the Euler characteristic of each open stratum of a stratification of σ − (0)which is induced from the canonical stratification of a divisor with normal crossings. Inthis case σ is obtained by taking first the blow-up X ′ → X = C n along the origin of C n ,5nd then taking the base change of an embedded resolution of ( P n − , Z ) by the projection X ′ → P n − where Z := P ( D ). The Euler characteristic of an open stratum is calculatedfrom those of the closed strata contained in the closure of the given stratum. So theassertion follows by induction on n using [DP] together with the embedded resolutionof ( P n − , Z ) obtained by blowing up along the proper transforms of all the edges of Z .Indeed, any intersection of the proper transforms of exceptional divisors can be written asa product of embedded resolutions for certain induced arrangements, see loc. cit. and [BS],Prop. 2.7 in this case. (If we blow up along only the proper transforms of dense edges, wecan not apply an inductive arguments since there is a problem as below: For two denseedges L ⊂ L ′ of Z ⊂ P n − , L is not necessarily a dense edge of the induced arrangementin L ′ .) This finishes the proof of Proposition (1.3). The following question arises naturally:
Question (Q).
Let D be an indecomposable essential central hyperplane arrangement in C n defined by a polynomial f of degree d . Then, is − n/d a pole of Z topf, ( s )?We have a positive answer to this question if n = 2 or n = 3 and D is reduced, seePropositions (1.5) and (1.8) below. Recently, W. Veys informed us that the answer isnegative in general, more precisely, if n = 3 with D non-reduced or n = 5 with D reduced,see Appendix.Assume, for example, n = 2 and d = P ei =1 m i with m i = mult D i D . Then(1 . . Z topf, ( s ) = 1 d s + 2 (cid:16) − e + e X i =1 m i s + 1 (cid:17) . This immediately follows from the definition of the zeta function since the embeddedresolution is obtained by one blow-up and 2 − e is the Euler characteristic of the openstratum in P . So − /d is a pole of order 2 if and only if 2 m i = d for some i . If − /d isnot a pole of order 2, then the coefficient C − /d of ds +2 is given by C − /d = 2 − e + e X i =1 dd − m i . The next Proposition gives a positive answer to Question (Q) in (1.4) for n = 2 where D may be non-reduced. This is a special case of [Ve3], Prop. 2.8. (W. Veys [Ve3]). With the above notation, assume n = 2 . Then − /d is a pole of Z topf, ( s ) . More precisely, if − /d is not a pole of order , then C − /d > if { } is a good dense edge of D , and C − /d < otherwise.Proof. See [Ve3], Prop. 2.8.
Assume n = 3 , and D is reduced. Let ν m ( m ≥ be the number ofpoints of Z := P ( D ) with multiplicity m . Then Z topf, ( s ) = 1 ds + 3 χ ( P \ Z ) + χ ( Z \ Z sing ) s + 1 + X m (cid:16) − m + ms + 1 (cid:17) ν m ms + 2 ! . n particular, − /d is the only candidate for the pole of order of Z topf, ( s ) , and is really apole of order if and only if d/ ∈ Z and ν d/ = 0 . If − /d is not a pole of order , thecoefficient C − /d of ds +3 is given by C − /d = 9 d − (cid:16) d − X m =2 d/ m ( m − d − m ν m (cid:17) . Proof.
Since the embedded resolution of ( P , Z ) is obtained by blowing up along thesingular points of Z , the first assertion follows from the definition of Z topf, ( s ) using thepartition of the summation over m = 2 and m = 2. This implies the second assertion sincethe coefficient of the double pole is given up to a nonzero multiplicative constant by2 − m + mdd − a − a − = 0 , where m = 2 a with a := d/ ∈ Z . For the simple pole case, we have C − /d = χ ( P \ Z ) + χ ( Z \ Z sing ) dd − X m =2 d/ (cid:16) − m + mdd − (cid:17) ν m d d − m . Here χ ( P \ Z ) = 3 − d + P m ( m − ν m ,χ ( Z \ Z sing ) = 2 d − P m m ν m . Indeed, the first equality is reduced to the calculation of χ ( Z ) which is obtained by usingthe short exact sequence 0 → Q Z ι ֒ → L i Q Z i → Coker ι → , since the cokernel of ι is supported on the singular points of Z and its rank at p is m p − m p is themultiplicity of Z at p .Substituting these, we see that C − /d is given by3 − d + 2 d d − X m =2 d/ ν m (cid:16) m − − mdd − − m ) d d − m + md ( d − d − m ) (cid:17) . After some calculation this is transformed to9 d − (cid:16) d − X m =2 d/ m ( m − d − m ν m (cid:17) . (The detail is left to the reader.) This finishes the proof of Proposition (1.6). A strong form of the conjecture in [DL] predicts that the multiplicity ofeach root of the zeta function is at most that of the b -function. In general, the multiplicityof the root − b -function of a reduced essential central hyperplane arrangement is n (see [Sa2], Th. 1), and this settles the problem for the root −
1. However, the problem7s rather difficult for the roots with multiplicity 2 even in the case n = 3. In this case theonly such root is − /d with d/ ∈ N and ν d/ = 0 by Proposition (1.6), but it is not easyto calculate the b -function. (Indeed, the multiplicity is calculated only in the case ν m = 0for m > D is reduced and n = 3. W. Veys has informed us that hehad verified an analogue of it for the (finer) motivic or Hodge zeta functions. (Here ‘finer’means that the non-vanishing of the pole for these do not imply that for Z topf, ( s ) althoughthe converse is true.) Let D be an indecomposable essential central hyperplane arrangementof degree d in C . Assume D is reduced. Then − /d is a pole of Z topf, ( s ) . More precisely,if − /d is not a pole of order , then the coefficient C − /d of ds +3 is strictly positive if { } is a good dense edge of D , i.e. if m < d/ for any m with ν m = 0 , and C − /d is strictlynegative otherwise.Proof. We may assume n = 3 since the case n = 2 is trivial. We may further assume { } is not a good dense edge of D , since the assertion in the good dense edge case easilyfollows from Proposition (1.6). We may thus assume ν m = 0 for some m := 2 a + e with0 < e < a := d/ a ∈ Z . Since the sum of the multiplicities ofany two singular points of Z is at most d + 1, we have ν m = 1 and m ≤ a − m + 1 = a − e + 1 for any m = m with ν m = 0 . By Proposition (1.6) the assertion C − /d < a + e )(2 a + e − e > a −
1) + X m ≤ a − e +1 m ( m − a − m ν m . To show the last inequality, we may replace m ( m − a − m with m ( m − a + e − , since a + e − ≥ a − m for m ≤ a − e + 1. Using (cid:0) d (cid:1) = P m (cid:0) m (cid:1) ν m , the assertion is then reduced to(2 a + e )(2 a + e − e > a −
1) + 3 a (3 a − − (2 a + e )(2 a + e − a + e − , i.e. (2 a + e )(2 a + e − a + 2 e − − a − a + e − e = 2(2 a + e − a − e )( a − e − > . Here a > e + 1, i.e. m = 2 a + e < d − D is indecomposable. So the assertion isproved. 8 . Calculation of b -functions2.1. For a nonconstant polynomial f ∈ K [ x , . . . , x n ] with char K = 0, the b -function b f ( s ) can be defined to be the minimal polynomial of the action of s on D n [ s ] f s / D n [ s ] f s +1 . This implies that b f ( s ) is invariant by extensions of K and its roots are rational numberssince the last assertion holds for K = C by [Ka1].Let i f : X ֒ → X × A K denote the graph embedding of f where X = A nK . Thenvia the global section functor, R f [ s ] f s is identified with the direct image by i f of the D X -module O X [ f ] in the notation of the introduction. This is compatible with extensionsof K . Moreover, the regular holonomic D X -module Gr αV (( i f ) ∗ O X [ f ]) corresponds via theglobal section functor to Gr αV ( R f [ s ] f s ), and via the de Rham functor to the λ -eigenspaceof Deligne’s nearby cycle sheaf ψ f C X ([De]) with λ = e − πiα if K = C , see [Ka2], [Ma2].This implies that the union of the eigenvalues of the Milnor monodromies for x ∈ f − C (0) ⊂ C n is independent of the choice of an embedding K ֒ → C since the α arerational numbers. b -functions of homogeneous polynomials. Assume that X = C n and f is ahomogeneous polynomial. Let F f denote the Milnor fiber of f , and H n − ( F f , C ) λ bethe λ -eigenspace of the Milnor cohomology by the action of the monodromy T , where n = dim X . Set ⌊ α ⌋ = max { k ∈ Z | k ≤ α } , e ( α ) = exp(2 πiα ) for α ∈ Q . By [Sa1], Th. 2, there is a decreasing filtration P on H n − ( F f , C ) λ such that(2 . . b f ( − α ) = 0 if Gr ⌊ n − α ⌋ P H n − ( F f , C ) e ( − α ) = 0 , where P coincides with e P in loc. cit. since f is homogeneous.Set U := P n − \ Z with Z := f − (0) ⊂ P n − . By [Sa1], Prop. 4.9, the filtration P on H n − ( F f , C ) λ is induced by the pole order filtration P on the meromorphic extension L ( k ) of a local system L ( k ) of rank one on U such that(2 . . H j ( U, L ( k ) ) = H j ( F f , C ) λ , where λ = exp( − πik/d ) with d = deg f . Here the local system L ( k ) is defined by thedecomposition π ∗ C F f = P d − k =0 L ( k ) , where π is the canonical projection from the affine Milnor fiber F f := f − (1) ⊂ C n onto U ⊂ P n − , and the action of the monodromy is the multiplication by exp( − πik/d ) on L ( k ) so that (2.2.2) holds, see [CS] or [Di], Prop. 6.4.6. Since P n − is simply connected, thelocal system L ( k ) is determined by the monodromies around the irreducible components Z j Z . These are given by the multiplication by exp(2 πim j k/d ) where m j is the multiplicityof the divisor Z along Z j .We can identify locally L ( k ) with O Y ( ∗ Z ) h − k/d as a D Y -module if h defines locally Z ⊂ Y := P n − . Then the pole order filtration P on L ( k ) is defined by(2 . . P i L ( k ) = O Y h − kd − i if i ≥ , and 0 otherwise.Note that the residue of the logarithmic connection on P i L ( k ) at a general point of Z j isthe multiplication by(2 . . (cid:0) − kd − i (cid:1) m j . The filtration P i = P − i on H n − ( U, L ( k ) ) = H n − ( F f , C ) λ is induced by P n − − i on L ( k ) using the de Rham complex L ( k ) → L ( k ) ⊗ O Y Ω Y → · · · → L ( k ) ⊗ O Y Ω n − Y , since the latter has the filtration P i = P − i defined by P − i L ( k ) → P − i L ( k ) ⊗ O Y Ω Y → · · · → P n − − i L ( k ) ⊗ O Y Ω n − Y . We have also the Hodge filtration F on L ( k ) such that F i L ( k ) ⊂ P i L ( k ) , and the Hodge filtration F on H n − ( U, L ( k ) ) = H n − ( F f , C ) λ is induced by the aboveformula with P replaced by F . L ( k ) . From now on, assume D = f − (0) is acentral hyperplane arrangement in C n . Let D i ( i = 1 , . . . , e ) be the irreducible componentsof D with multiplicity m i . Then Z = P ( D ) ⊂ P n − and Z i = P ( D i ). Let D nnc denotethe smallest subset of D such that D \ D nnc is a divisor with normal crossings. Set Z nnc = P ( D nnc ) ⊂ P n − . Note that d = deg f = P ei =1 m i .For k ∈ { , . . . , d − } and I ⊂ { , . . . , e − } with | I | = k −
1, define(2 . . α Ii = (cid:26) − m i k/d if i / ∈ I ∪ { e } ,1 − m i k/d if i ∈ I ∪ { e } . α IL = P D i ⊃ L α Ii . Σ I = { p ∈ Z nnc \ Z e | α Ip = 0 } , where L is an edge of D , and we set α Ip := α IL if P ( L ) = { p } . (See Remark 3.6 (iii) belowfor another way of the definition of the α Ii .) Here it should be noted that in order to applythe theory in [ESV] (and also in [STV]), we must have a regular singular connection on a trivial line bundle, i.e. the following condition should be satisfied:(2 . . P ei =1 α Ii = 0 . d = P ei =1 m i . Note also that α Ie is used in an essentialway for (2.3.4) below (i.e. the condition of [STV]) although it does not appear in thedefinition of the connection on the affine space C n − which is given below.For i ∈ { , . . . , e − } , let e i = dg i /g i with g i a linear function defining Z i \ Z e in P n − \ Z e ∼ = C n − . Set ω I := P e − i =1 α Ii e i . It defines a connection ∇ ω I on O U (where U = P n − \ Z ) such that ∇ ω I u = du + uω I for u ∈ O U . The corresponding local system is isomorphic to L ( k ) by comparing their local monodromiesas remarked in (2.2). Consider the de Rham cohomology H • DR ( U, ( O U , ∇ ω I )), which iscalculated by the complex of rational forms (Ω • U ( U ) , ∇ ω I ) since U is affine. Set A p = P i < ··· for any nonzero dense edges L ⊂ D . Assume D is reduced (i.e. m i = 1) and ( k, d ) = 1. Then condition(2.3.4) is satisfied for any I with | I | = k − α IL / ∈ Z for any nonzero edge L .Moreover, this assumption implies that ψ f,λ C X , the nearby cycle sheaf with eigenvalue λ := exp( − πik/d ), is supported at the origin. (Indeed, in case the last assertion isnot true, there is d ′ ∈ (0 , d ) and k ′ ∈ N such that k/d = k ′ /d ′ . This follows from thecalculation of the Milnor cohomology in (2.2) to x ∈ D \ { } . Here the degree d ′ of thedefining equation of D at x ∈ D \ { } becomes strictly smaller. But this contradicts theassumption ( k, d ) = 1.) The above assertion implies further the vanishing of the lowerMilnor cohomology H j ( F f , C ) λ for j < n −
1, since the nearby cycle sheaf ψ f,λ C X is aperverse sheaf up to the shift of complex by n −
1. If moreover D is indecomposable, thenwe get the nonvanishing of the highest Milnor cohomology H n − ( F f , C ) λ by (2.2.2), sincethe indecomposability is equivalent to the nonvanishing of the Euler characteristic χ ( U ),see [STV]. 11ote that Theorem 4.2(e) in [Sa2] remains valid in the non-reduced case as follows. Let V ( I ) ′ be the subspace of A n − generated by e J := e j ∧ · · · ∧ e j n − forany J = { j , . . . , j n − } ⊂ I with j < j < · · · < j n − . Let V ( I ) be the image of V ( I ) ′ in H n − ( A • , ω I ∧ ) , where ω I and α I = ( α Ii ) are as in (2 . . Assume V ( I ) = 0 and (2 . . holds. Then b f ( − kd ) = 0 .Proof. By (2.2.1) it is enough to show that the image of e J by the injection ι n − I in (2.3.3)is contained P L ( k ) ⊗ O Y Ω n − Y in the notation of (2.2). Here P n H n − ( F f , C ) λ = 0 since P − L ( k ) = 0. By definition the image of a ∈ A = C by ι I is a global section v a of a free O Y -submodule L I of L ( k ) such that the residue of the connection at the generic point of Z i is the multiplication by α Ii in (2.3.1). Set Z I ∪{ e } := S nk =1 Z j k with j n := e . Then v a ⊗ e J ∈ L I ( Z I ∪{ e } ) ⊗ Ω n − Y , since e J ∈ Ω n − Y (log Z I ∪{ e } ) = Ω n − Y ( Z I ∪{ e } ). Thus the assertion is reduced to L I ( Z I ∪{ e } ) ⊂ P L ( k ) , and this is shown by comparing (2.2.4) and (2.3.1). Indeed, the eigenvalue of the residueof the connection on L I ( Z I ∪{ e } ) is shifted by − Z j for j ∈ I ∪ { e } ,but it is not smaller than − m j k/d even after this shift by (2.3.1). So Theorem (2.5) isproved. In case (i), n/d is a jumping coefficientby Teitler’s refinement [Te] of Mustat¸ˇa’s formula [Mu] for multiplier ideals using only denseedges. Hence it is a root of b f ( s ) up to a sign by [ELSV].In case (iii), condition (2.3.4) is satisfied for any I with | I | = n − k = n and( n, d ) = 1, see Remark (2.4) above. By [FT], Example 4.5, the highest degree cohomologyof the Aomoto complex H n − ( A • , ω I ∧ ) has a monomial basis (independently of I ) underthe genericity condition on D d . Take a subset I = { i , . . . , i n − } ⊂ { , . . . , d − } , such that the corresponding form e I = e i ∧· · ·∧ e i n − is a member of the obtained monomialbasis. Since (2.3.4) is satisfied, the image of e I in the cohomology of the local system doesnot vanish. So the assertion follows from Theorem (2.5) (i.e. [Sa2], Th. 4.2(e)). (i) In the above argument, the image of e I by ι n − I is independent of thechoice of I up to a nonzero constant multiple. Indeed, the injection ι I in (2.3.3) is definedby using the trivial line bundle L I in the proof of Theorem (2.5) which is determined bythe eigenvalues α Ii in (2.3.1). If we take another I ′ ⊂ { , . . . , d − } with | I ′ | = n − e I ′ = 0, then, using the trivialization given by L I , a nonzero constant section of L I ′ isidentified with the rational function c g I ′ /g I where c ∈ C ∗ and g I = Q i ∈ I g i in the notationof (2.3). This gives the difference between ι jI and ι jI ′ for any j . So the independence followssince g I e I = c ′ g I ′ e I ′ with c ′ ∈ C ∗ . 12ii) We can also identify the image of e I by ι n − I with an element of the Gauss-Maninsystem of f . The problem is then closely related to the torsion of the Brieskorn lattice.
3. The rank 3 case
In this section we assume n = 3 and give two proofs of the case (ii) in Theorem 2. Notethat the case n ≤ From now on we assume n = k = 3 . We will write p ⊂ i if { p } ⊂ Z i , and set α Ip = α IL if P ( L ) = { p } .In the notation of (2.3.1) we will study the following three conditions:( a ) α Ip / ∈ Z > for any p ∈ Z nnc = P ( D nnc ).( b ) ∃ p ∈ (cid:0)S i ∈ I Z i (cid:1) sing \ Z e .( c ) Z \ ( Z e ∪ Σ I ∪ { p } ) is connected. (i) In the case n = 3, condition ( a ) coincides with condition (2.3.4) whichimplies that the inclusion (2.3.3) is a quasi-isomorphism. Note that we have always theinequality of the dimensions, see [LY], Prop. 4.2.(ii) For i, j, k ⊃ p , there is a well-known relation(3 . . e i ∧ e j = e i ∧ e k − e j ∧ e k , which is easily checked by setting g i = x , g j = y and g k = x + y . This also follows from therelations of the Orlik-Solomon algebra which are given by ∂ ( e i ∧ e j ∧ e k ) for i, j, k ⊃ p , seee.g. [OT], p. 60. As in [BDS], Lemma 1.4, this implies for η = P e − i =1 β i e i and p ∈ Z nnc \ Z e (3 . .
2) If π p ( ω I ∧ η ) = 0, then α Ip β i = β p α Ii for any i ⊃ p .Here β p = P i ⊃ p β i , and π p ( ω I ∧ η ) is the p -component in the direct sum decomposition in[BDS], 1.3.2 H ( U, Q ) = L p L p , where p runs over ( Z red ) sing \ Z e , and L p is a vector space of rank m ′ p − m ′ p themultiplicity of Z red at p . More precisely L p has a basis consisting of e i ∧ e k with i ⊃ p and i = k where k is any fixed member such that k ⊃ p . This also follows from the definitionof the Orlik-Solomon algebra mentioned after (3.2.1), see e.g. [OT], p. 60.We also get(3 . .
3) If p ∈ ( Z i ∩ Z j ) \ ( Z nnc ∪ Z e ), then α Ii β j = α Ij β i .13n case α Ii = 0 (i.e. m i = d/
3) for any i ∈ I , we have by (3.2.2–3)(3 . .
4) If π p ( ω I ∧ η ) = 0 and p / ∈ Σ I , then β i /α Ii is independent of i ⊃ p .(iii) Lemma 1.4 in [BDS] or above (3.2.2) is essentially known to the specialists, see[LY], Lemma 3.1 (and also [Fa], [Li2], [Yu]). Here the situation is localized at p , i.e.the lines not passing through p are neglected, by using the fact that the relations of theOrlik-Solomon algebra are of the form ∂ ( e J ) for certain J and are compatible with thedecomposition by p . With the notation and the assumption of (2 . , assume n = k = 3 andthere is I ⊂ { , . . . , e − } such that | I | = 2 and conditions ( a ) , ( b ) and ( c ) in (3 . aresatisfied. Then b f ( − /d ) = 0 where f is a defining polynomial of D .Proof. Let p be as in condition ( b ) in (3.1), and assume the following condition is satisfied: π p ( ω I ∧ η ) = 0 for any p = p .Then η is a multiple of ω I , i.e. β i /α Ii is independent of i , see Remark (3.2)(ii). So we canapply Theorem (2.5) (i.e. [Sa2], Th. 4.2(e)), and conclude that b f ( − /d ) = 0. This finishesthe proof of Proposition (3.3). We may assume that { } is not a good dense edge,since we can apply the case (i) otherwise. By Proposition (3.3), it is sufficient to show thefollowing: Assertion.
There is an irreducible component Z e of Z together with a subset I ⊂{ , . . . , e − } such that | I | = 2 and conditions ( a ), ( b ) and ( c ) in (3.1) are satisfiedchanging the order of { , . . . , e } if necessary.Note first that α Ip can be an integer only in the case d/ ∈ Z . (Indeed, we have m p := P i ⊃ p m i < d , and hence α Ip ≡ m p /d Z unless d/ ∈ Z .) Then the aboveassertion is shown in the case d/ / ∈ Z as follows.Since α Ip / ∈ Z for any p ∈ Z nnc , condition (a) is trivially satisfied and Σ I = ∅ for anychoice of I . Assuming D central and indecomposable, there is p ∈ Z sing together with Z e and I satisfying condition (b). As for condition (c), it is not satisfied only in the casethere is Z i passing through p and such that Z i ∩ Z i ′ ⊂ ( Z i ∩ Z e ) ∪ { p } for any i ′ / ∈ { i, e } .(Otherwise, for any Z i passing through p , there is Z i ′ such that Z i ∩ Z i ′ Z e ∪ { p } .) Inthis case every Z i passes through either p or Z i ∩ Z e . This implies that | Z nnc | = 2 since D is indecomposable. Then, replacing Z e with Z i containing Z nnc , we may take p to beany point of Z sing \ Z nnc and I is chosen so that { p } = T i ∈ I Z i . Thus the assertion isproved in this case.We may now assume a := d/ ∈ Z . Since { } is not a good dense edge, there is p ∈ Z nnc with multiplicity > a . On theother hand, we may assume that there is p ∈ Z nnc with α Ip ∈ Z , i.e. its multiplicity14s divisible by a , since otherwise the above conditions are easily satisfied. Thus we mayassume that there are p , p ∈ Z nnc with multiplicity 2 a + 1 and a respectively and hence Z nnc = { p , p } , since d = 3 a . So the assertion is proved by the same argument as above. It is also possible to prove Theorem 2(ii) bytaking p to be the point with multiplicity m p > d , which exists since we may assumethat { } is not a good dense edge as in (3.4). In this case there is a line Z d − which isdifferent from the line at infinity Z d and does not contain p since D is indecomposable.Moreover there are at least two lines Z , Z passing through p such that their intersectionswith Z d − are ordinary double points of Z and furthermore their intersections with Z d donot have multiplicity a so that conditions ( a ) and ( b ) in (3.1) are satisfied by setting I = { , } . Indeed, we have m p > d , d − m p ≥
2, and hence d >
6, and moreover thenumber of lines Z i such that i ⊃ p and Z i ∩ Z d − is an ordinary double point of Z is atleast m p − − ( d − − m p ) > , since | S i ⊃ p Z i ∩ Z d − | ≥ m p −
1. So the condition on the intersection with Z d − issatisfied. For the intersection with Z d we can exclude the case where a point of Z hasmultiplicity a since this case has a very special structure as explained at the end of (3.4)(e.g. the singular points of Z other than this point and p are ordinary double points) sothat we can easily choose Z , Z satisfying the above conditions in this case.We can then prove Theorem 2(ii) without using Proposition (3.3) but using (3.2.1).Indeed, by Theorem (2.5) (i.e. [Sa2], Th. 4.2(e)), it is enough to show(3 . .
1) If ( P i α Ii e i ) ∧ ( P j β j e j ) = c e ∧ e for some c ∈ Q , then c = 0.Under the assumption of (3.5.1) we get by using (3.2.1)(3 . . α Ip β i = β p α Ii if i ⊃ p and i > α Ip = 0 since m p > d . So we may assume(3 . . β i = 0 if i ⊃ p and i > β i with β i − c ′ α Ii for any i where c ′ := β p /α Ip . (Note that this change of β i does not affect the hypothesis of (3.5.1).) Since m p >
4, (3.5.2) and (3.5.3) imply β + β = β p = 0 . On the other hand, by (3.2.3) applied to the intersections of Z , Z with Z d − , we get β /α I = β d − /α Id − = β /α I , where α I = α I = 0 and α Id − = 0 since Z is reduced. So β = β = 0, and (3.5.1) follows. (i) It does not seem easy to generalize the above arguments to the non-reduced case. If p is taken to be the point with the highest multiplicity, there is anexample as follows: Assume a >
6, and let f = ( xy ( x − y )) a − ( x + y − z )( x + y − z )( x + 2 y − z )(2 x + y − z ) z . d = 3 a , and there does not exist I such that the argument in (3.5) can be applied if weset p = (0 , , Z i ( i = 1 , . . . ,
8) denote the lines defined by the linear factorsof f respecting the order of the factors, where e = 8. Here Z e must be the line defined by z = 0 since conditions ( a ) and ( b ) in (3.1) cannot be satisfied otherwise. Then the singularpoints of Z \ ( { p } ∪ Z e ) contained in Z or Z have all multiplicity a , and moreover Z red has multiplicity 3 at these points. So the argument in (3.5) cannot be applied.(ii) For a more complicated example, we might consider the following: Let E be anelliptic curve in the dual projective space P , and G be the subgroup of torsion points oforder three. This defines a projective hyperplane arrangement in P with e = | G | = 9,see e.g. [Li]. Let G be a subgroup of G with order 3. Assume a >
6. To the linescorresponding to the elements of G we give the multiplicity a −
2, while the other lineshave multiplicity 1. Then d = 3 a , and I ∪ { e } should correspond to G + p ⊂ G for some p ∈ G in order to satisfy condition ( a ) in (3.1). (Indeed, if there are g , g ∈ I ∪ { e } such that their images in G/G are different, then there is g ∈ G such that the imagesof g , g , g in G/G are all different and moreover g + g + g = 0. The last conditionis equivalent to the condition that the three lines corresponding to g , g , g intersect atone point. Then condition ( a ) is not satisfied at this point.) So p is contained in Z e ,and hence condition ( b ) cannot be satisfied. Thus we cannot prove a generalization ofTheorem 2 in this case by using Theorem (2.5) (i.e. the generalization of [Sa2], Th. 4.2(e)to the nonreduced case).(iii) In order to apply the theory in [ESV] and [STV], we have to choose the residues α i of the connection satisfying the two conditions (2.3.2) and (2.3.4). In our case we have α i = n i − m i k/d with n i ∈ Z by the monodromy condition, and P i n i = k since P i m i = d .Then, to satisfy (2.3.2), an easy way is to choose a subset J of { , . . . , e } with | J | = k andset n i = 1 for i ∈ J and n i = 0 otherwise. Here there are two possibilities depending onwhether e ∈ J or e / ∈ J . Since e corresponds to the divisor at infinity, this makes somedifference in the calculation of the Aomoto complex which is defined on the complementaffine space C n − . In (2.3.1) we considered the former case where I = J \ { e } . However, itis also possible to consider the latter case where I = J so that | I | = k instead of | I | = k − α Ii = (cid:26) − m i k/d if i / ∈ I ,1 − m i k/d if i ∈ I .In the latter case, however, it is usually more difficult to satisfy the three conditions in(3.1).(iv) If n = 3, d ≤ p Z = 3 for any p ∈ Z nnc in the notation of (2.3), the b -function of a reduced hyperplane arrangement is calculated in [Sa2].16 ppendix by Willem Veys University of Leuven, Department of MathematicsCelestijnenlaan 200 B, B-3001 Leuven (Heverlee), Belgium
This appendix describes some examples solving Question (Q) in (1.4) negatively. I thankthe authors of this paper for writing some details.
A.1. Example . We first explain an example of a nonreduced hyperplane arrangementwith n = 3, d = 9. Let f = xy ( x − y ) z ( x − z ) . This gives a negative answer to Question (Q) in (1.4). Indeed, we have χ ( U ) = 1 by usingthe affine space defined by x = 0. We have χ ( Z ◦ i ) = − x = 0, and the Euler characteristic is 0 for the latter. So we get Z topf, ( s ) = 19 s + 3 − s + 1 − s + 1 − s + 1 + (cid:16) − s + 1 (cid:17) s + 2+ (cid:16) − s + 1 + 12 s + 1 + 14 s + 1 (cid:17) s + 2 + 2 s + 1 (cid:16) s + 1 + 14 s + 1 (cid:17)! . Set Φ( s ) = (9 s + 3) Z topf, ( s ). Since s +1 + s +1 vanishes by substituting s = − /
3, we getΦ( − /
3) = 1 − (cid:16) − (cid:17) − (cid:16) − (cid:17) = 0 . So the pole at − / p = (0 : 1 : 0) ∈ P ,the line at infinity is { y = 0 } , and I corresponds to the two lines with multiplicities 2 and4. This assertion is also shown by a calculation using the computer program Asir. A.2. Example . There is an example of a reduced hyperplane arrangement with n = 5and d = 10, giving a negative answer to Question (Q) in (1.4), and which is defined by apolynomial f as below: f = ( x − y )( x − y )( x − y )( x − y )( x − y )( x + y + z ) zuv ( u + v + z ) . In fact, let Z , Z , Z be closed subvarieties of Y := P defined by Z = { x = y = z = 0 } , Z = { u = v = z = 0 } , Z = { x = y = 0 } . Let ρ : Y ′ → Y be the composition of the blow-up of Y along Z , Z and the blow-upalong the proper transform of Z . This gives an embedded resolution of ( Y, Z ) where Z := { f = 0 } ⊂ Y . We have a partition { S i } i =0 ,..., of Y = P defined by S = { z = 0 } , S i = Z i ( i = 1 , , S = { z = 0 } \ ( Z ∪ Z ) . S ′ i := ρ − ( S i ) ( i = 0 , . . . , . Let x ′ , y ′ , u ′ , v ′ be affine coordinates of S defined respectively by xz , yz , uz , vz . Then S ′ = e C x ′ ,y ′ × C u ′ ,v ′ , S ′ = e P x,y,z × P u,v , S ′ = P u,v,z × P x,y , where e C x ′ ,y ′ and e P x,y,z are respectively the blow-up of C x ′ ,y ′ and P x,y,z along (0 ,
0) and(0 : 0 : 1). Here the lower indices x,y etc. indicate the coordinates. Note that each S i is aunion of strata of the stratification associated to the divisor with normal crossings ρ − ( Z ).So we get Z topf, ( s ) = X i =0 Ψ i ( s )10 s + 5 , where Ψ i ( s ) / (10 s +5) is the sum of the factors of Z topf, ( s ) associated to the strata containedin S ′ i . Since the stratification is compatible with the above product structure, we getΨ ( s ) = (cid:16) − s + 1 + 5( s + 1) + (cid:16) − s + 1 (cid:17) s + 2 (cid:17) · (cid:16) − s + 1 + 3( s + 1) (cid:17) , Ψ ( s ) = 17 s + 3 (cid:16) − s + 1 + 11( s + 1) + (cid:16) − s + 1 (cid:17) s + 2 (cid:17) · (cid:16) − s + 1 (cid:17) , Ψ ( s ) = 14 s + 3 (cid:16) − s + 1 + 6( s + 1) (cid:17) · (cid:16) − s + 1 (cid:17) , Ψ ( s ) = 0 . Indeed, let Z ′ be the divisor on P x,y,z defined by the product of linear factors of f whichare linear combinations of x, y, z , and similarly for Z ′′ with x, y replaced by u, v . Then χ ( P \ Z ′ ) = 4 , χ ( P \ Z ′′ ) = 1 , χ ( Z ′ \ Sing Z ′ ) = − , χ ( Z ′′ \ Sing Z ′′ ) = − , and the number of ordinary double points of Z ′ and Z ′′ are respectively 11 and 6. Thecalculation for P u,v and P x,y is similar, and we get the formulas for Ψ ( s ) and Ψ ( s ) sincethe definition of Ψ ( s ) , Ψ ( s ) is compatible with the above product structure using theformula: χ ( X × X ) = χ ( X ) · χ ( X ) for topological spaces X , X . As for the first terms,note that the codimensions of the centers Z , Z are 3, and the multiplicities of f at thegeneric points of Z and Z are respectively 7 and 4. The term (cid:0) − s +1 (cid:1) s +2 comesfrom the exceptional divisor of the blow-up along the proper transform of Z , where themultiplicity of f at the generic point of Z is 5 and Z has codimension 2.The argument is similar for Ψ ( s ). Here the Euler number of the smooth part and thenumber of ordinary double points change since the varieties are restricted to (the blow-upof) the affine space C . The vanishing of Ψ ( s ) follows from the C ∗ -action on S ′ = S compatible with the stratification, which is defined by λ ( x : y : u : v ) = ( λ x : λ y : u : v )for λ ∈ C ∗ . 18ubstituting s = − to the above formulas, we getΨ (cid:0) − (cid:1) = − · , Ψ (cid:0) − (cid:1) = − · · , Ψ (cid:0) − (cid:1) = 17 · , and hence the pole of Z topf, ( s ) at s = − vanishes. For the moment it is not clear whether − is a root of b f ( s ). References [ACLM1] Artal Bartolo, E., Cassou-Nogu`es, P., Luengo, I. and Melle Hern´andez, A., Monodromyconjecture for some surface singularities, Ann. Sci. Ecole Norm. Sup. (4) 35 (2002), 605–640.[ACLM2] Artal Bartolo, E., Cassou-Nogu`es, P., Luengo, I. and Melle Hern´andez, A., Quasi-ordinarypower series and their zeta functions, Mem. Amer. Math. 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