Monodromy Conjecture for log generic polynomials
aa r X i v : . [ m a t h . AG ] J u l MONODROMY CONJECTURE FOR LOG GENERIC POLYNOMIALS
NERO BUDUR AND ROBIN VAN DER VEER
Abstract.
A log generic hypersurface in P n with respect to a birational modification of P n is by definition the image of a generic element of a high power of an ample linear serieson the modification. A log very-generic hypersurface is defined similarly but restrictingto line bundles satisfying a non-resonance condition. Fixing a log resolution of a product f = f . . . f p of polynomials, we show that the monodromy conjecture, relating the motiviczeta function with the complex monodromy, holds for the tuple ( f , . . . , f p , g ) and for theproduct f g , if g is log generic. We also show that the stronger version of the monodromyconjecture, relating the motivic zeta function with the Bernstein-Sato ideal, holds for thetuple ( f , . . . , f p , g ) and for the product f g , if g is log very-generic. Introduction
Let F = ( f , . . . , f p ) be a tuple of polynomials f i ∈ C [ x , . . . , x n ]. Let f = Q pi =1 f i . The topological zeta function of F Z topF ( s , . . . , s p )is a rational function, cf. Definition 3.1. We denote the polar locus of this rational function,that is, the support of the divisor of poles, by P ( Z topF ).On the other hand, one has the monodromy support of F S F ⊂ ( C ∗ ) p , cf. Definition 3.2. If p = 1, this is the set of all eigenvalues of the monodromy on thecohomology of the Milnor fibers of f . Let Exp : C p → ( C ∗ ) p be the map α exp(2 πiα )coordinate-wise. Conjecture 1.1 (Monodromy Conjecture) . Let F be a tuple of polynomials in C [ x , . . . , x n ] .Then Exp ( P ( Z topF )) ⊂ S F . A stronger conjecture involves the
Bernstein-Sato ideal B F , the ideal of generated by b ∈ C [ s , . . . , s p ] satisfying b p Y i =1 f s i i = P p Y i =1 f s i +1 i for some P ∈ D [ s , . . . , s p +1 ], where D is the ring of linear algebraic differential operators on C n . When p = 1, the monic generator of this ideal is the b -function of f . Let Z ( B F ) denotethe zero locus of B F in C p . It was recently proven in [7] that Exp ( Z ( B F )) = S F , Mathematics Subject Classification.
Key words and phrases.
Monodromy conjecture; motivic zeta function; Milnor fibration; Bernstein-Satoideal. xtending the case p = 1 due to Malgrange, Kashiwara. Conjecture 1.2 (Strong Monodromy Conjecture) . Let F be a tuple of polynomials in C [ x , . . . , x n ] . Then P ( Z topF ) ⊂ Z ( B F ) . For p = 1, the conjectures are the analog due to Denef-Loeser [8] of a classical conjecturefor p -adic local zeta functions of Igusa [15].Among the known cases with p = 1 of the stronger conjecture are: plane curves [19],tame hyperplane arrangements [25]. Among the known cases with p = 1 of the weakerversion are: hyperplane arrangements [6], non-degenerate surfaces [17] and non-degeneratethreefolds [11]. See the survey [22] for more cases.For p >
1, the Monodromy Conjecture was posed by Loeser, cf. [23]. It is known fortuples of plane curves [23], and of hyperplane arrangements [4].For p >
1, the Strong Monodromy Conjecture was posed in [4]. It is known for tuplesfactorizing a tame hyperplane arrangement [2], and for tuples of linear polynomials [26].In this note we address both conjectures in presence of two notions genericity with respectto birational modifications.
Setup 1.3.
We fix a non-zero polynomial f ∈ C [ x , . . . , x n ] and maps Y µ (cid:15) (cid:15) (cid:31) (cid:127) / / ¯ Y ¯ µ (cid:15) (cid:15) C n (cid:31) (cid:127) / / P n such that: • the bottom map is the inclusion of the complement of the hyperplane at infinity, Y = ¯ µ − ( C n ), and µ = ¯ µ | Y ; • ¯ µ is a composition of blowing ups of smooth closed subvarieties; • ¯ µ is a log resolution of the divisor div( f ) in P n of the rational function f , that is, theunion of the exceptional locus of ¯ µ with div( f ) is a simple normal crossings divisorin Y .Starting with f , one can always reach such a setup. Having fixed this set-up, we maketwo definitions. The first one is: Definition 1.4.
We say that a statement holds for log generic polynomials in C [ x , . . . , x n ]if: - for every ample line bundle L on ¯ Y , and- for all k ≫ g ∈ C [ x , . . . , x n ], where g is a defining polynomial for the imageunder µ of the restriction to Y of a generic member of the finite dimensional space | L ⊗ k | .For the second definition we will restrict to very general line bundles, that is, elements ofa certain non-empty subcone A vg of the integral ample cone of ¯ Y as in Definition 5.2. Definition 1.5.
We say that a statement holds for log very-generic polynomials in C [ x , . . . , x n ]if: - for every ample line bundle L on ¯ Y such that L ∈ A vg , and for all k ≫ g ∈ C [ x , . . . , x n ], where g is a defining polynomial for the imageunder µ of the restriction to Y of a generic member of the finite dimensional space | L ⊗ k | . Remark 1.6. (i) If a statement holds log generically then it holds log very-generically.(ii) The morphism µ is also a log resolution of f g for any log generic polynomial g , byBertini Theorem. Moreover µ is a minimal log resolution for g , in the sense that if µ factorsthrough another log resolution µ ′ of f , then µ ′ is not a log resolution of g .(iii) Even if f = 1, log generic polynomials can be highly singular depending on the logresolutions chosen.(iv) Log generic polynomials g can be obtained from generic elements of symbolic powersof ideals as follows. If L ≃ ¯ µ ∗ ( O P n ( d )) ⊗ O ¯ Y O ¯ Y ( − A )for some positive integer d and some effective relatively-ample divisor A supported on theexceptional locus of ¯ µ , then ¯ µ ∗ ( L ⊗ k ) ≃ J ( k ) ( kd )where J is the ideal subsheaf µ ∗ ( O ¯ Y ( − A )) of O P n , and J ( k ) = µ ∗ ( O ¯ Y ( − kA )) is the k -thsymbolic power of J . Then g are generic elements for k ≫ P n , J ( k ) ( kd )) → Γ( C n , J ( k ) ( kd )) . (v) Log generic polynomials not necessarily non-degenerate polynomials even in the case f = 1, although there is an analogy. Non-degenerate polynomials are generic elements infinite-dimensional vector spaces, generated by monomials, of polynomials with fixed Newtonpolytope. However, by taking log resolutions of non-monomial ideals, one can generateexamples for which the image of the map (1) cannot be generated by monomials.(vi) The condition imposed on log generic polynomials to obtain log very-generic polyno-mials is an analog of the non-resonance condition for non-degenerate polynomials of [20].To state the results, we keep Setup 1.3 fixed. Theorem 1.7.
Let F = ( f , . . . , f p ) be a tuple of non-zero polynomials in C [ x , . . . , x n ] , and f = Q pi =1 f i . Then:(a) the Monodromy Conjecture for ˜ F = ( f , . . . , f p , g ) is true, for log generic polynomials g ;(b) the Strong Monodromy Conjecture for ˜ F = ( f , . . . , f p , g ) is true, for log very-genericpolynomials g . The same holds after taking products:
Theorem 1.8.
With the same setup:(a) the Monodromy Conjecture for the product f g is true, for log generic polynomials g ;(b) the Strong Monodromy Conjecture for the product f g is true, for log very-genericpolynomials g . The proof of Theorem 1.7 ( a ) relies on a formula for the monodromy zeta function associ-ated to a tuple of polynomials due to Sabbah [24], generalizing a classical result of A’Campo or p = 1. The monodromy zeta functions recover the monodromy support of a tuple ofpolynomials by [5], cf. Theorem 3.4. The case p = 1 of this fact was pointed out by Denefas a consequence of the perversity of the nearby cycles complex. Using this we show thatall the candidates for polar hyperplanes of Z top ˜ F arising from µ give components of the mon-odromy support of ˜ F . This also shows that the results in this note hold more generally formotivic zeta functions instead of topological zeta functions. Theorem 1.8 ( a ) is a corollaryof Theorem 1.7 ( a ).To address the parts ( b ) of the theorems, we show in Proposition 5.5 that every candidatepolar hyperplane of the relevant zeta functions is an actual polar hyperplane of order one.We prove then firstly Theorem 1.8 ( b ) by adapting Loeser’s proof from [20] that non-resonantcompact codimension 1 faces of the Newton polytope of the germ of a non-degenerate hyper-surface singularity give roots of the b -function. This method had also appeared in [18, 19] inthe 1-dimensional case. The main differences with [20] stem from the fact that we do not as-sume compactness of exceptional divisors. We use a criterion to produce roots of b -functionsdue to Hamm [14] slightly improving a result of Malgrange [21] used in [20]. Like [20], weuse a non-vanishing theorem for local systems of Esnault-Viehweg [10]. To stress that logvery-generic polynomials have complicated singularities, we show in 6.1 that the roots ofthe b -functions produced here are not necessarily negatives of jumping numbers. Theorem1.7 ( b ) follows from Theorem 1.8 ( b ) by using results for generalized Bernstein-Sato ideals ofGyoja [13] and Budur [4].In Section 2 we fix notation. In Section 3 we recall some facts about the objects of study.In Section 4 we prove parts ( a ) of Theorems 1.7 and 1.8. In Section 5 we prove parts ( b ) ofTheorems 1.7 and 1.8. Section 6 contains some remarks. Acknowledgement.
We thank L. Wu and P. Zhao for useful discussions. The first authorwould like to thank MPI Bonn for hospitality during writing this article. The first authorwas partly supported by the grants STRT/13/005 and Methusalem METH/15/026 fromKU Leuven, G097819N and G0F4216N from FWO (Research Foundation - Flanders). Thesecond author is supported by a PhD Fellowship from FWO.2.
Notation.
For the proofs of the main results, we have to introduce some notation. With fix Setup1.3. We let f = Q pi =1 f i with f i ∈ C [ x , . . . , x n ]. We set the following: • ¯ J exc is the set of irreducible components of the exceptional locus of ¯ µ ; • J exc = { W ∈ ¯ J exc | W ∩ Y = ∅} ; • L is a very ample line bundle on ¯ Y ; • H ∈ | L ⊗ k | is a general element and k > • g is a defining polynomial for µ ( H ∩ Y ) in C n ; • F = ( f , . . . , f p ); • ˜ F = ( f , . . . , f p , f p +1 ) with f p +1 = g ; • ˜ f = f · f p +1 ; ¯ J is the union of ¯ J exc and the set of irreducible components of support of the divisor¯ µ ∗ (div( ˜ f )) on ¯ Y ; • J = { W ∈ ¯ J | W ∩ Y = ∅} ; • W ◦ = W \ ∪ W ′ ∈ ¯ J \{ W } W ′ for W ∈ ¯ J ; • W ◦ J ′ = ( ∩ W ∈ J ′ W ) \ ( ∪ W ∈ ¯ J \ J ′ W ) for J ′ ⊂ ¯ J . • For every W ∈ ¯ J : ◦ n W = ord W ( K ¯ µ ) + 1, where K ¯ µ is the relative canonical divisor of ¯ µ ; ◦ a i,W = ord W ( f i ); ◦ a W = ord W ( f ) = P pi =1 a i,W ; ◦ N W = ord W ( ˜ f ) = P p +1 i =1 a i,W . We draw the attention that some of these notions depend on the integer k >
0, althoughthis has been suppressed from the notation.3.
Invariants of singularities
We introduce the objects that form the subject of our results. We keep the setup andnotation from Section 2. We note that µ is a log resolution of ˜ f and is an isomorphism over C n \ ˜ f − (0). The following invariant was introduced by Denef-Loeser: Definition 3.1.
The topological zeta function of ˜ F is Z top ˜ F ( s , . . . , s p +1 ) = X ∅6 = J ′ ⊂ J χ ( W ◦ J ′ ) Y W ∈ J ′ a ,W s + · · · + a p +1 ,W s p +1 + n W . The support in C p +1 of the divisor of poles of the rational function Z top ˜ F , the polar locus ,is a hyperplane arrangement and will be denoted by P ( Z top ˜ F ). The hyperplane { a ,W s + · · · + a p +1 ,W s p +1 + n W = 0 } is called the candidate polar hyperplane from the component W . Definition 3.2.
The monodromy support of ˜ F is the subset S ˜ F ⊂ ( C ∗ ) p +1 consisting of α ∈ ( C ∗ ) p +1 for which there exist a point x ∈ ˜ f − (0) with H ∗ ( U x , L α ) = 0 , where U x is the complement of ˜ f − (0) in a small open ball around x in C n , and L α is thelocal system obtained as the pullback under the restriction of ˜ F to U x of the rank one localsystem on ( C ∗ ) p +1 with monodromy α i around the i -th missing coordinate hyperplane.An equivalent definition of S ˜ F is that this the support of the generalized monodromyaction on the generalization of the nearby cycles complex [24], by [4, 5]. The monodromysupport S ˜ F is a finite union of torsion-translated codimension-one affine algebraic subtori of( C ∗ ) p +1 , by [5].To a point x ∈ ˜ f − (0) one associates the monodromy zeta function Z mon ˜ F ,x of the stalk at x of the generalized nearby cycles complex of ˜ F . We can take as definition the following ormula from [24, Proposition 2.6.2], [12, Th´eor´eme 4.4.1], which recovers a classical formuladue to A’Campo in the case p = 1: Theorem 3.3. Z mon ˜ F ,x ( t , . . . , t p +1 ) = Y W ∈ J ( t a ,W · · · t a p +1 ,W p +1 − − χ ( W ◦ ∩ µ − ( x )) . Denote by PZ ( Z mon ˜ F ,x )the support of the divisor on ( C ∗ ) p +1 associated to Z mon ˜ F ,x , the union of the zero and the polarlocus, each being a finite union of torsion-translated codimension-one algebraic subtori.Let Ω ⊂ ˜ f − (0) be a finite set consisting of general points of each stratum of a Whitneystratification of ˜ f − (0). By [5] we have: Theorem 3.4. S ˜ F = [ x ∈ Ω PZ ( Z mon ˜ F ,x ) . Log generic polynomials
In this section we address log generic polynomials. For the proof of Theorem 1.7 ( a ), weuse the following estimate on asymptotic topological Euler characteristics: Lemma 4.1.
Let ¯ Y be a smooth projective variety, L a very ample line bundle on ¯ Y , and V ⊂ ¯ Y a non-empty Zariski locally closed subset. Let H ∈ | L ⊗ k | be a generic element for k > . Then, for k ≫ , χ ( V \ H ) = ( − dim V deg L ( ¯ V top ) · k dim V + lower order terms in k, where ¯ V top is the union of the top-dimensional irreducible components of the closure of V . (If dim V = 0 , there are no “lower order terms in k ”, by convention.) In particular, for k ≫ , ( − dim V χ ( V \ H ) > . Moreover, χ ( V ∩ H ) = ( − dim V − deg L ( ¯ V top ) · k dim V + lower order terms in k if dim V > , and in general χ ( H \ V ) = ( − dim ¯ Y − deg L ( ¯ Y ) · k dim ¯ Y + lower order terms in k. Proof.
Assume first V is closed and irreducible. In this case, V ∩ H is also irreducible andcomplete, hence χ ( V ∩ H ) = Z c SM ( V ∩ H )where c SM denote the Chern-Schwartz-MacPherson class, see for example [1, 2.2]. By [1,Proposition 2.6], c SM ( V ∩ H ) = H H · c SM ( V ) . Hence χ ( V ∩ H ) = ( − dim V − k dim V Z c ( L ) dim V · c SM ( V ) dim V + lower order terms in k. y [1, Theorem 1.1], Z c ( L ) dim V c SM ( V ) dim V = deg L ( V ) > . Thus χ ( V ∩ H ) = ( − dim V − deg L ( V ) · k dim V + lower order terms in k. We will extend this result now to the case when V is closed but not necessarily irreducible.Let V i with i ∈ I be the irreducible components of V . Using inclusion-exclusion, we canwrite χ ( V ∩ H ) = X i ∈ I χ ( V i ∩ H ) + X W m W · χ ( W ∩ H )where W are irreducible components of intersections of at least two distinct irreduciblecomponents of V , and m W are suitable multiplicities independent of k since H is generic. Itfollows by the first part of this proof that χ ( V ∩ H ) = ( − dim V − X i ∈ I dim V i =dim V deg L ( V i ) · k dim V + lower order terms in k = ( − dim V − deg L ( V top ) · k dim V + lower order terms in k where V top is the union of the top-dimensional irreducible components of V .Now let V be locally closed and denote by ¯ V the closure of V in ¯ Y . Let Z = ¯ V \ V , sothat Z is closed in ¯ Y . Then χ ( V \ H ) = χ ( ¯ V \ H ) − χ ( Z \ H )= χ ( ¯ V ) − χ ( ¯ V ∩ H ) − χ ( Z ) + χ ( Z ∩ H )= ( − dim V deg L ( ¯ V top ) · k dim V + lower order terms in k, where the last equality follows from the case handled above. This proves the first assertion.Writing χ ( V ∩ H ) = χ ( ¯ V ∩ H ) − χ ( Z ∩ H ) , we obtain the second assertion, since dim( Z ∩ H ) < dim V if dim V >
1, and if dim V = 1then Z = ∅ by the genericity of H .Next, χ ( H \ V ) = χ ( H ) − χ ( ¯ V ∩ H ) + χ ( Z ∩ H )= ( − dim ¯ Y − deg L ( ¯ Y ) · k dim ¯ Y + lower order terms in k, as claimed. (cid:3) ( a ) . We use the notation from Section 2 and take k ≫
0. Weshow that the Monodromy Conjecture holds for the tuple ˜ F = ( f , . . . , f p +1 ).It is enough to show that every candidate polar hyperplane for Z top ˜ F arising from µ ismapped via the exponential map into the monodromy support of ˜ F . That is, that Exp ( { a ,W s + . . . + a p +1 ,W s p +1 + n W = 0 } ) ⊂ S ˜ F for every W ∈ J . y Theorem 3.4, it is thus enough to show that for every W ∈ J , the locus ( p +1 Y i =1 t a i,W i = 1 ) ⊂ ( C ∗ ) p +1 is contained in PZ ( Z mon ˜ F ,x ) for some x ∈ Ω.For W ∈ J and x ∈ Ω, let W ◦ x = W ◦ ∩ µ − ( x ) ,J x = { W ∈ J | W ◦ x = ∅} . Then(2) Z mon ˜ F ,x ( t , . . . , t p +1 ) = Y W ∈ J x ( t a ,W · · · t a p +1 ,W p +1 − − χ ( W ◦ x ) by Theorem 3.3.If W = H and x ∈ Ω is a general point on µ ( H ∩ Y ), the vector a • ,W in Z p +1 is equal to(0 , . . . , ,
1) and χ ( W ◦ ∩ µ − ( x )) = χ ( { x } ) = 1 . Moreover, J x = { H } in this case. Thus Z mon ˜ F ,x = ( t p +1 − − and so { Q p +1 i =1 t a i,W i = 1 } = PZ ( Z mon ˜ F ,x ), which proves the claim in this case.For the remaining cases fix x ∈ Ω a general point of a stratum of a Whitney stratificationof f − (0). It is enough to show that the locus { Q p +1 i =1 t a i,W i = 1 } is contained in PZ ( Z mon ˜ F ,x )for every W in J x \ { H } . For such W , let W x = ( W \ ∪ W ′ ∈ ¯ J \{ H } W ′ ) ∩ µ − ( x ) , so that W ◦ x = W x \ H. Then χ ( W ◦ x ) = ( − dim W ◦ x deg L (( W x ) top ) · k dim W x + lower order terms in k, by Lemma 4.1, where ( W x ) top is the union of the top-dimensional irreducible componentsof the Zariski closure of W x , and there are no “lower order terms in k ” if dim W x = 0.In particular, χ ( W ◦ x ) = 0 for k ≫
0, and hence every W ∈ J x \ { H } contributes to theright-hand side of (2) with a non-trivial factor before cancellations.Suppose that a non-trivial irreducible factor P ( t ) of Q p +1 i =1 t a i,W i − W ∈ J x \ { H } cancels out from (2) and the zero locus of P ( t ) does not lie in PZ ( Z mon ˜ F ,x ). Let J ′ ⊂ J x \ { H } be the set of all W ∈ J x \ { H } with strictly positive multiplicity of P ( t ) as a factor of Q p +1 i =1 t a i,W i −
1. Since the latter polynomial is reduced, this multiplicity has to equal 1. Thecancellation then implies X W ∈ J ′ χ ( W ◦ x ) = 0 . Let r = max { dim W ◦ x | W ∈ J ′ } . Then 0 = X W ∈ J ′ , dim W ◦ x = r ( − r deg L (( W x ) top ) · k r + lower order terms in k, here for r = 0 there are no “lower order terms in k ”. For k ≫ >
0, and hence the coefficient of k r is non-zero. ( a ) . We let f p +1 = g and ˜ f = f · f p +1 as in the proof ofTheorem 1.7 ( a ), for k ≫
0. Since Z top ˜ f ( s ) = Z top ˜ F ( s, . . . , s ), the restriction of the polar locusof Z top ˜ F to the line s = . . . = s p +1 = s contains the polar locus of Z top ˜ f . The conclusion thenfollows from Theorem 1.7 ( a ) and the fact that the restriction of the monodromy support S ˜ F of ˜ F to s = . . . = s p +1 = s equals the monodromy support S ˜ f of ˜ f , by [5, Theorem 2.11]. Log very-generic polynomials
In this section we address log very-generic polynomials. With fix Setup 1.3.
Since ¯ µ is a composition of blowing ups ofsmooth closed subvarieties, Z ⊕ M W ∈ ¯ J exc Z [ W ] ∼ −→ Pic( ¯ Y ) , ( d, b W ) ¯ µ ∗ O P n ( d ) ⊗ O ¯ Y − X W ∈ ¯ J exc b W W is an isomorphism of finitely generated abelian groups. We let A ⊂
Pic( ¯ Y ) be the subset ofample isomorphism classes. Then A ⊂ R + ⊕ M W ∈ ¯ J exc R + [ W ]and A is a subcone , that is, if L ∈ A then every integral point in the ray R + L belongs to A ,where R + denote the strictly positive real numbers.We introduce the subcone A vg of A used in Definition 1.5 of log very-general polynomials: Definition 5.2.
Let A vg ⊂ A be the set of isomorphism classes of ample line bundles on ¯ Y such that for each W ∈ J exc n W b W b W ′ Z for all W ′ ∈ ¯ J exc \ { W } with W ∩ W ′ = ∅ .Note that the condition defining A vg in A is actually a condition on the ¯ µ -ample cone,which coincides with the image of the projection of A to the space indexed by ¯ J exc . Lemma 5.3.
The subset A vg of A is a non-empty subcone.Proof. By definition A vg is a subcone if non-empty. We show that it is non-empty. Fix L ∈ A ample with associated coordinates b W for W ∈ ¯ J exc .Choose integers p W ≫ W ∈ ¯ J exc such that n W p W /p W ′ Z for all pairs ( W, W ′ )of different elements in ¯ J exc . This is possible since ¯ J exc is finite. Moreover, L − X W ∈ ¯ J exc p W W is an ample Q -divisor class by [16, Example 1.3.14]. Let p = Q W ∈ ¯ J exc p W . Thus p ( L − X W ∈ ¯ J exc p W W ) s an ample integral divisor class. Replacing L by this new divisor class, one replaces b W by p ( b W + 1 /p W ) for each W ∈ ¯ J exc . Moreover, n W p ( b W + 1 /p W ) p ( b W ′ + 1 /p W ′ ) = p W n W p W b W + 1 · p W ′ b W ′ + 1 p W ′ is not an integer since p W ′ does not divide the numerator. This proves the claim. (cid:3) We keep the notation from Section 2 and let k ≫ F = ( f , . . . , f p +1 ), with f = Q pi =1 , ˜ f = f f p +1 , and f p +1 = g is log generic. Proposition 5.5. If f p +1 = g is log very-generic, then:(i) Every candidate polar hyperplane of Z top ˜ F ( s , . . . , s p +1 ) arising from the exceptionallocus of µ is a polar hyperplane of order one.(ii) Every candidate pole of Z top ˜ f ( s ) arising from the exceptional locus of µ is a pole oforder one.Proof. We prove ( ii ) first. We have(3) Z top ˜ f ( s ) = X ∅6 = J ′ ⊂ J χ ( W ◦ J ′ ) Y W ∈ J ′ N W s + n W . Moreover, N W = a W + a p +1 ,W where a p +1 ,W = ord W ( µ ∗ ( µ ( H ∩ Y ))) . On the other hand, H ∩ Y = µ ∗ ( µ ( H ∩ Y )) − X W ∈ J exc a p +1 ,W ( W ∩ Y )as a divisor on Y , and O Y ( H ∩ Y ) ≃ L ⊗ k | Y by definition of H . Let L ≃ ¯ µ ∗ O P n ( d ) ⊗ O ¯ Y − X W ∈ ¯ J exc b W W be the unique representation of the isomorphism class of L in the cone A . Then(4) a p +1 ,W = kb W for W ∈ J exc , W = H, in which case a W = 0 , W ∈ J \ ( J exc ∪ { H } ) . From now on we will now restrict to those L in A vg as in Definition 5.2, this being thereason why we prove the proposition only log very-generically and not log generically.We show that the candidate pole from W ∈ J exc , − n W N W = − n W a W + kb W , is a pole of order one of Z top ˜ f ( s ) for k ≫ irstly, the pole order is at most 1. If the order would be >
1, then from formula (3) wesee that there must exist W ′ ∈ J \ { W } such that n W N W = n W a W + kb W = n W ′ N W ′ . This is impossible for W ′ J exc for large k . If this equality happens for W ′ ∈ J exc forinfinitely many k ∈ N , by taking limit as k goes to infinity we obtain that n W b W b W ′ = n W ′ which is excluded by the condition that L is in A vg .Since the pole order is at most 1, to show that the order is equal to 1 it is enough to showthat the evaluation at s = − n W /N W of( N W s + n W ) Z top ˜ f ( s )is a non-zero number for k ≫
0. By (3), this number is X ∅6 = J ′ ⊂ J \{ W } χ ( W ◦ J ′ ∪{ W } ) Y W ′ ∈ J ′ N W N W n W ′ − N W ′ n W . The denominators are all different than zero, as we have seen already. For k ≫
0, using (4)and the asymptotic behaviour from Lemma 4.1 for the Euler characteristics, the dominantterm corresponds to J ′ = { H } and it is equal to ( − n − deg L ( ¯ Y ) k n . Since this term ispositive, this proves (ii).Now we show ( i ). Let W ∈ J exc . The candidate polar hyperplane for Z top ˜ F from W is { P p +1 i =1 a i,W s i + n W = 0 } . Note that N W = P p +1 i =1 a i,W . Since Z top ˜ f ( s ) = Z top ˜ F ( s, . . . , s ), therestriction of the polar locus of Z top ˜ F to the line s = . . . = s p +1 = s contains the polar locusof Z top ˜ f ( s ). By part (ii), W contributes with the pole − n W N W = ( n W + p +1 X i =1 a i,W s i = 0 )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s = ... = s p +1 = s to Z top ˜ f ( s ), and we have seen that our assumptions imply that n W N W = n W ′ N W ′ for every W ′ ∈ J \ { W } for k ≫
0. Thus the polar locus of Z top ˜ F must contain the candidate from W . (cid:3) We fix, as always in section, the setup and nota-tion from Section 2 and take k ≫
0. We will show:
Proposition 5.7. If f p +1 = g is log very-generic:(i) Every polar hyperplane of Z top ˜ F ( s , . . . , s p +1 ) arising from the exceptional locus of µ isan irreducible component of Z ( B ˜ F ) .(ii) Every pole of Z top ˜ f ( s ) arising from the exceptional locus of µ is a root of the b -functionof ˜ f . Granted this proposition, we can complete: roof of parts ( b ) of Theorems 1.7 and 1.8. The non-exceptional components contribute triv-ially with irreducible components of the zero locus of the Bernstein-Sato ideal (resp. withroots of the b -function), by localizing around general point, hence smooth, on such a com-ponent. Thus the claim follows from the previous proposition. (cid:3) The rest of the section is dedicated to the proof of the proposition.
Proof of Proposition 5.7 (ii).
The proof takes a few steps. Let W ∈ J exc . We prove that − n W /N W is a root of the b -function of ˜ f for k ≫ W ∩ Y is compact, one can apply [20, 6.6] directly. However, W ∩ Y is typically notcompact, so we have to adapt the proof of [20, 6.6].Let α ∈ R . The multi-valued form ˜ f α dx ∧ . . . ∧ dx n gives a global section in Γ( U, Ω nU ⊗ C L U ) where U = C n \ ˜ f − (0), Ω nU is the sheaf of holomorphic n -forms, and L U is the rank one local system on U defined as the pullback via ˜ f of the rankone local system on C ∗ with monodromy multiplication by exp( − πiα ) around the origin.By construction of ˜ f , µ is an isomorphism over U , V := Y \ ( ˜ f ◦ µ ) − (0) ∼ −→ U. We have ω = µ ∗ ( ˜ f α dx ∧ . . . ∧ dx n ) ∈ Γ( V, Ω nV ⊗ C L V )where L V = µ − L U . For every W ′ ∈ ¯ J , the order of vanishing of ω along W ′ is well-definedand(5) ord W ′ ( ω ) = n W ′ − αN W ′ . The monodromy of L V around W ′ isexp( − πi · ord W ′ ( ω )) = exp( − πiαN W ′ ) . Moreover, ω has a meromorphic extension to ¯ Y across the simple normal crossings divisor A = ¯ Y \ V = X W ′ ∈ ¯ J W ′ . More precisely, ω ∈ Γ( ¯
Y , Ω n ¯ Y (log A ) ⊗ O ¯ Y M ) ⊂ Γ( V, Ω nV ⊗ C L V )where M = O ¯ Y − X W ′ ∈ ¯ J (ord W ′ ( ω ) + 1) · W ′ ! is defined (a definition is necessary since the coefficients are in R ) as M = L canY ⊗ O ¯ Y O ¯ Y − X W ′ ∈ ¯ J ⌊ ord W ′ ( ω ) + 1 ⌋ · W ′ ! , with L canV the canonical Deligne extension of L V , and ⌊ ⌋ denoting the round-down. Recallthat L canV is a line bundle on ¯ Y extending O V ⊗ C L V and is defined as follows. Around a eneral point of W ′ , let z be a local holomorphic function on ¯ Y defining W ′ , and let u be alocal multi-valued frame for L V . Then L canV = O ¯ Y − X W ′ ∈ ¯ J { ord W ′ ( ω ) + 1 } · W ′ ! is defined by declaring z { ord W ′ ( ω )+1 } u to be a local holomorphic frame, that is, locally L canV ≃ O ¯ Y · z { ord W ′ ( ω )+1 } u, where { } denotes the fractional part.By definition of M , the M -valued log differential form ω has no poles nor zeros on V .Therefore ω induces an isomorphism of invertible sheaves O ¯ Y ∼ −→ Ω n ¯ Y (log A ) ⊗ O ¯ Y M , and hence an isomorphism M − ∼ −→ Ω n ¯ Y (log A ) . Let us denote the residue of ω along W by η ∈ Γ( W, Ω n − W (log A W ) ⊗ O ¯ Y M )where A W = ( A − W ) | W so that W ◦ = W \ A W . By definition, η is locally ( zω/dz ) | W ◦ where z is a holomorphicfunction on ¯ Y defining W .We take from now α = − n W N W . The effect of this choice is that ord W ( ω ) + 1 = 0 . This implies that η = 0 and(6) c ( L canV | W ) = − X W ′ ∈ ¯ J W { ord W ′ ( ω ) + 1 } · [ W ′ | W ] ∈ H ( W, R )where ¯ J W := { W ′ ∈ ¯ J | W = W ′ and W ∩ W ′ = ∅} . Equation (6) guarantees that there exists a rank one local system L on W ◦ with monodromyaround W ′ | W with W ′ ∈ ¯ J W precisely − exp(2 πi { ord W ′ ( ω ) + 1 } ) , by applying for example [3, Theorem 1.2 and § L to W is L can = L canV | W . Thus η ∈ Γ( W, Ω n − W (log A W ) ⊗ O W M ) ⊂ Γ( W ◦ , Ω n − W ◦ ⊗ C L ) . That is, η is a meromorphic L -twisted differential form with no poles nor zeros on W ◦ , and M| W is the smallest invertible sheaf on W with this property. This is the first ingredientneeded to apply [10]. ince H ∩ W is an irreducible component of A W and is a very ample divisor class on W for k ≫
0, one has thatΩ n − W (log A W ) ≃ O W ( K W + A W ) ≃ O W ( B ) ⊗ O W L ⊗ k is nef and big for k ≫ B = K W + A W − ( H ∩ W ). This isthe second ingredient needed to apply [10].We now assume further that L ∈ A vg as in Definition 5.2, this being the reason why weprove the proposition only log very-generically and not log generically. This and (4) implyfor k ≫ W ′ ∈ ¯ J W , n W N W N W ′ Z , or, equivalently by (5), ord W ′ ( ω ) Z . Thus none of the monodromies of L is 1. This is the third and last ingredient needed toapply [10].We can now apply the main theorem of [10] and obtain that the form η determines anon-zero class in H n − ( W ◦ , L ). Since W \ ( H ∩ W ) is affine, its subset W ◦ is also affine. Itfollows by [9, (1.5)] and its proof that H n − ( W ◦ , L ) = H n − c ( W ◦ , L ) . Therefore there existsa cycle γ ∈ H n − ( W ◦ , L ∨ )with coefficients in the dual local system of L , such that Z γ η = 0 . From now on all the arguments are as in the proof of [20, 6.6], except at the last step aswe will point out. Consider a Gelfand-Leray form˜ f − nWNW dx ∧ . . . ∧ dx n d ˜ f on U . A local computation shows that the pullback by µ ∗ extends over W ◦ and µ ∗ ˜ f − nWNW dx ∧ . . . ∧ dx n d ˜ f !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W ◦ = η up to multiplication by a non-zero constant.Let N be the lowest common multiple of all N W ′ for W ′ ∈ ¯ J . Let ˜ Y → C be thenormalization of the base change of ˜ f ◦ µ : Y → C by the morphism C → C , t t N . Let˜ W ◦ be the inverse image of W ◦ in ˜ Y . Then the natural map ν : ˜ W ◦ → W ◦ is ´etale and ν ∗ L is the constant sheaf. Thus ν ∗ ( η ) ∈ H n − ( ˜ W ◦ , C ) and one has a cycle˜ γ ∈ H n − ( ˜ W ◦ , C ) such that Z ˜ γ ν ∗ η = 0 . ince ˜ W ◦ is smooth, ˜ f lifts to a trivial fibration on a small tubular neighborhood T of ˜ W ◦ in ˜ Y . Let T t be the fibers for small t . By parallel transport, ˜ γ = ˜ γ (0) for a horizontal family˜ γ ( t ) ∈ H n − ( T t , C )for small t . Pushing forward to C n , we obtain a horizontal multi-valued family of cycles γ ( t ) ∈ H n − ( ˜ f − ( t ) , C )for small t = 0, such that lim t → t − nWNW Z γ ( t ) dx ∧ . . . ∧ dx n d ˜ f exists and is a non-zero constant.The last statement implies directly that − n W /N W is a root of the b -function of ˜ f byapplying [14, III]. This finishes the proof of Proposition 5.7 (ii). (cid:3) Proof of Proposition 5.7 ( i ) . Let W ∈ J exc . Since k ≫
0, we have a W,p +1 > n W . Denote by L the set of all l = ( l , . . . , l p +1 ) ∈ Z p +1 > such that n W l a W, + · · · + l p +1 a W,p +1 ( l a W ′ , + · · · + l p +1 a W ′ ,p +1 ) Z for all W ′ ∈ ¯ J \ { W } with W ′ ∩ W = ∅ . Then for all ( l , . . . , l p ) ∈ Z p> and for l p +1 ≫ l , . . . , l p ), ( l , . . . , l p +1 ) ∈ L . To see this, one can use (4), the constraint from Definition 5.2, and take limits as l p +1 → ∞ .It follows from the proof of Proposition 5.7 (ii) that for l ∈ L , r W, l := − n W l a W, + · · · + l p +1 a W,p +1 is a root of the b -function of f l . . . f l p +1 p +1 . This easily implies that for all b ( s , . . . , s p +1 ) ∈ B l ˜ F , b ( l r W, l , . . . , l p +1 r W, l ) = 0 , cf. [4, Lemma 4.20]. Here B l ˜ F is the generalized Bernstein-Sato ideal consisting of b ∈ C [ s , . . . , s p +1 ] such that b p Y i =1 f s i i = P p +1 Y i =1 f s i + l i i for some P ∈ D [ s , . . . , s p +1 ].Denote q W, l := ( l r W, l , . . . , l p +1 r W, l ) ∈ C p +1 , so that we can write Q := { q W, l | l ∈ L } ⊂ [ l ∈ L Z ( B l ˜ F ) =: Z . Notice that all q W, l lie on the polar hyperplane of Z top ˜ F ( s , . . . , s p +1 ) contributed by W , L := { a W, s + · · · + a W,p +1 s p +1 + n W = 0 } . Denote by t i : C p +1 → C p +1 with i = 1 , . . . , p + 1, the maps t i ( c , . . . , c p +1 ) = ( c , . . . , c i − , c i − , c i +1 , . . . , c p +1 ) . hen we recall from [4, Proposition 4.10] that we can write the zero locus of B l ˜ F as Z ( B l ˜ F ) = p +1 [ i =1 l i − [ j =0 t l p +1 p +1 . . . t l i +1 i +1 t ji Z ( B e i ˜ F )where e i is the i -th standard basis vector.Now take a small ball B around (0 , . . . , , − n W /a W,p +1 ) ∈ C p +1 . It follows from thedescription of Z ( B l ˜ F ) above and the definition of Z that only finitely many irreducible com-ponents of Z intersect B . Denote the reducible variety obtained by taking the union of thesecomponents by Q . Then Q is an algebraic Zariski closed subset of C p +1 . Since Q ∩ B ⊂ Q , bytaking Zariski closure in C p +1 we also have Q ∩ B ⊂ Q . Lemma 5.8 shows that Q ∩ B = L .We thus conclude that L ⊂ Q ⊂ Z . We can then find a vector d ∈ L such that L ⊂ Z ( B d ˜ F ) = p +1 [ i =1 d i − [ j =0 t d p +1 p +1 . . . t d i +1 i +1 t ji Z ( B e i ˜ F ) . (7)Suppose that in (7), the hyperplane L is contained in t jp +1 Z ( B e p +1 ˜ F ) for some j >
0. Apply t − jp +1 to the inclusion L ⊂ t jp +1 Z ( B e p +1 ˜ F ) to find that { a W, s + · · · + a W,p +1 s p +1 + n W − ja W,p +1 = 0 } ∈ Z ( B e p +1 ˜ F ) . But then n W − ja W,p +1 <
0, which contradicts the main result of [13]. Similarly we find thatfor all i = 1 , . . . , p and j ≥ L t d p +1 p +1 . . . t d i +1 i +1 t ji Z ( B e i ˜ F ) . We thus conclude that L ⊂ Z ( B e p +1 ˜ F )However, from the definition of generalized Bernstein-Sato ideals we have Z ( B e p +1 ˜ F ) ⊂ Z ( B ˜ F ) , so this finishes the proof. (cid:3) Lemma 5.8.
With the notation of the proof of Proposition 5.7 ( i ) , Q ∩ B = L .Proof. Fix an arbitrary l ′ = ( l , . . . , l p ) ∈ Z p> . We already remarked that for l p +1 ≫ l = ( l , . . . , l p +1 ) ∈ L . For such sufficiently large l p +1 we have moreover that q W, l ∈ Q ∩ B .Keeping l ′ fixed, but letting l p +1 get larger, all the points q W, l lie on the same line T l ′ , whichhas parametric representation T l ′ = (cid:26)(cid:18) , . . . , , − n W a W,p +1 (cid:19) + t (cid:18) l , . . . , l p , − a W, l + · · · + a W,p l p a W,p +1 (cid:19) | t ∈ C (cid:27) . We conclude that all lines of this form are contained in Q ∩ B . It follows that L ∩ ( Q p> × Q ) ⊂ Q ∩ B. Since the left hand side is clearly Zariski dense inside L , we conclude that L ⊂ Q ∩ B . Theother inclusion is immediate since Q ∩ B ⊂ L . (cid:3) . Last remarks
Even in the simplest situations, the roots of the b -function ofthe log very-generic polynomials that we produced in this note are not necessarily jumpingnumbers [16, Definition 9.3.22], although small jumping numbers give roots of the b -function[16, 9.3.25].Let µ : Y → C be the composition of the blow up at the origin, followed by the blow up ofpoint on the exceptional divisor. Let g a log very-generic polynomial on C , as in Definition1.5, where we fit µ into the Setup 1.3 by taking f = 1 and extending µ to ¯ µ : ¯ Y → P trivially. Let W and W be the exceptional irreducible divisors of µ , which in this case arethe same as those of ¯ µ . Let α i = n W i N W i = ord W i ( K ¯ µ ) + 1ord W i g ( i = 1 , . Then both − α and − α are roots of the b -function of g by Proposition 5.7 (ii). However,either α or α is a jumping number of g , but not both at the same time.More precisely, let A rel = { ( b , b ) ∈ Z | b W + b W is µ -ample } be the integral relatively-ample cone. Then A rel is the image of the projection of the amplecone A of ¯ Y to the space with coordinates indexed by W and W , cf. 5.1. One has A rel = { ( b , b ) ∈ N | < b < b < b } by applying the numerical relative-ampleness criterion. Let A vgrel be the projection to A rel ofthe subcone A vg of Definition 5.2. Then A vgrel = A rel \ { ( b , b ) ∈ A rel | b = 2 b } . To a log very-generic polynomial g one attaches a largely-scaled point ( b , b ) in A vgrel , thecoordinates corresponding to the power L ⊗ k from which g was generated. Then α i = i +1 b i . Thus one can define for i = 1 , , a subcone A i of A vgrel corresponding to the log very-genericpolynomials g having α i as jumping number. A short computation reveals that A = { ( b , b ) ∈ A vgrel | b < b } = A vgrel \ A . Theorems 1.7 and 1.8 are still valid if g is replaced by a power g l for any integer l >
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KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium, and BCAM, Mazarredo 14,48009 Bilbao, Spain.
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