BERNSTEIN-SATO POLYNOMIALS

(LECTURE NOTES, UPC BARCELONA, 2015)

NERO BUDUR

Abstract. These are lecture notes based on a series lectures at the Summer school Mul-tiplier ideals, test ideals, and Bernstein-Sato polynomials, UPC Barcelona, September 7-10,2015. The main scope of these notes is to give some answers to the questions: What is thegeometry behind the Bernstein-Sato polynomials ? What can you do with them?

Contents

21. Classical Bernstein-Sato polynomials 2

1.1. Origins. 21.10. Proof of existence. 41.18. Example: hyperplane arrangements 61.25. The geometry behind classical Bernstein-Sato polynomials. 7

2. Bernstein-Sato polynomials of ideals and varieties 82.1. Bernstein-Sato polynomials for ideals. 82.8. Bernstein-Sato polynomials of schemes. 92.11. Example: generic determinantal varieties. 92.17. Relation with multiplier ideals. 10

3. V -filtration 113.1. Riemann-Hilbert correspondence 123.4. V -filtrations on D-modules. 133.10. Bernstein-Sato polynomials of sections of D-modules. 153.13. The geometry behind the V -filtration. 15

4. Bernstein-Sato ideals 184.1. Bernstein-Sato ideals for many functions 184.8. Ideals of Bernstein-Sato type. 204.14. Relation with local systems. 224.18. Proof of Part (a) of Theorem 4.5 23

References 25

Date: September 6, 2015.This work was partially supported by an FWO grant, a KU Leuven OT grant, and a Flemish Methusalem

grant.1

These are lecture notes from a series of lectures at the Summer school Multiplier ideals,test ideals, and Bernstein-Sato polynomials, UPC Barcelona, September 7-10, 2015. Theycorrect and go further than an earlier version used for a series of lecture at the Summerschool Algebra, Algorithms, and Algebraic Analysis, Rolduc Abbey, Netherlands, September2-6, 2013.

The main scope of these notes is to give some answers to the questions: What is thegeometry behind the Bernstein-Sato polynomials ? What can you do with them?

We can only cover a few topics here. The choice is biased and reflects personal taste. Fora better view on where the topics covered here fit and for a more complete set of references,see the surveys [Bu-12] and [BW-15a].

We thank L. Saumell San Martin for some corrections.

1. Classical Bernstein-Sato polynomials

1.1. Origins. There are historically two different sources which lead to the classical Bernstein-Sato polynomial: matrix theory and generalized special functions theory.

Proposition 1.2. (Cayley) Let f = det(xij) be the determinant of a n×n matrix (xij)1≤i,j≤nof indeterminates. Then

(s+ 1)(s+ 2) . . . (s+ n)f s = det(∂/∂xij)fs+1.(1.2.1)

Generalizations by M. Sato lead to the theory of prehomogeneous vector spaces, see [Ki-03].While the main goal is classification, functional relations such as (1.2.1) are of fundamentalimportance. A prehomogeneous vector space is a vector space V over a field K of charac-teristic zero with an algebraic action ρ : G → GL(V ) of an algebraic group G such thatit admits a Zariski open orbit U ⊂ V . A semi-invariant is a rational function f ∈ K(V )such that f(ρ(g)x) = χ(g)f(x) for some character χ : G → K∗, for all g ∈ G and x ∈ V .The irreducible components of the complement V \U are given by homogeneous irreduciblepolynomials which are semi-invariants. Moreover, all semi-invariants are of this type. WhenK = C and G is a complex reductive group, the dual action ρ∗ : g 7→ tρ(g)−1 makes(G, V ∗, ρ∗) into a prehomogeneous vector space as well. One can show that for a semi-

invariant f of (G, V, ρ) associated to the character χ, f ∗(y) = f(y) is a semi-invariant of(G, V ∗, ρ∗) associated to χ−1.

Proposition 1.3. (M. Sato) Let (G, V, ρ) be an n-dimensional complex prehomogeneousvector space with G. If f is a semi-invariant of degree d, there exists a non-zero polynomialb(s) of degree d such that

b(s)f(x)s = f ∗(∂/∂x1, . . . , ∂/∂xn)f(x)s+1.

Example 1.4. Let G = GL(n,C) act on the space V = Mn(C) of complex n × n matricesvia the usual multiplication ρ(g) : x 7→ gx. Then (G, V, ρ) is a prehomogeneous vector space,f(x) = det(x) is a semi-invariant for the character χ : G→ C∗ given by χ(g) = det(g), andProposition 1.3 generalizes Proposition 1.2.

Definition 1.5. Let f ∈ K[x1, . . . , xn] be a polynomial with coefficients in a field K ofcharacteristic zero. The Bernstein-Sato polynomial of f , also called the b-function, is the

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non-zero monic polynomial bf (s) of minimal degree among those b ∈ K[s] such that

b(s)f s = Pf s+1(1.5.1)

for some operator P ∈ K[x, ∂/∂x, s].

Remark 1.6. T. Oaku [Oa-97] has used the theory of Grobner bases to develop and imple-ment the first algorithm for computing Bernstein-Sato polynomials. Nowadays, the computerprograms Macaulay2, Risa/Asir, Singular, have packages and commands allowing thecomputation of Bernstein-Sato polynomials.

Remark 1.7. One can define Bernstein-Sato polynomials of a (germ of an) analytic functionf : Cn → C by replacing the ring of polynomials with the ring of (germs of) analytic func-tions, and then allowing the operator P to have analytic coefficients instead of polynomialcoefficients in x.

Remark 1.8. It is non-trivial that non-zero Bernstein-Sato polynomials exist. The exis-tence was proved by I.N. Bernstein, independently of Sato’s proof for semi-invariants ofprehomogeneous vector spaces.

Bernstein’s work was motivated by a question I.M. Gelfand posed in 1963: what is themeaning of f s, the complex power of a polynomial? More precisely, let f ∈ R[x1, . . . , xn]and s ∈ C. For Re(s) > 0 define a locally integrable function on Rn

f s+(x) =

{f(x)s if f(x) > 0,

0 if f(x) ≤ 0.

Then the question is if f s+ admits a meromorphic continuation to all s ∈ C and, if so, todescribe the poles. This was positively answered by M. Atiyah and Bernstein-Gelfand whodescribed the poles in terms of a resolution of singularities of f . A more precise result wasproved by Bernstein:

Proposition 1.9. As a distribution, f s+ admits a meromorphic continuation with poles inthe set A− N, where A is the set of roots of bf (s).

Proof. As a distribution, f s+ is defined by its value on smooth compactly supported functionsφ,

〈f s+, φ〉 =

∫Rn

φ(x)f s+dx,

which converges and defines a holomorphic distribution for Re(s) > 0. Now, for Re(s) > 0,

b(s)

∫Rn

φ(x)f s+dx =

∫f>0

φ(x)b(s)f sdx

=

∫Rn

φ(x)(P (s)f s+1)+dx,

where b(s)f s = P (s)f s+1 as in Definition 1.5. If P (s) =∑

β aβ(x, s)(∂∂x

)β, define the adjoint

operator

P (s)∗ =∑β

(−1)|β|(∂

∂x

)βaβ(x, s).

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Integrating by parts we obtain∫Rn

φ(x)(P (s)f s+1)+dx =

∫Rn

P (s)∗(φ(x))f s+1+ dx.

So, for Re(s) > 0,

〈f s+, φ〉 =1

b(s)〈f s+1

+ , P (s)∗(φ)〉.

The right-hand side is well-defined and holomorphic on {s | Re(s) > −1}\b−1(0). Thiscontinues meromorphically the left-hand side to {s | Re(s) > −1} with poles in the zerolocus of b(s). By iterating this process, we obtain the Proposition. �

1.10. Proof of existence. We now sketch the proof of the existence of a non-zero polyno-mial as in Definition 1.5. This will be a crash course on the basic theory of D-modules. Formore details see [Bj-79]. The proof of the existence of a non-zero Bernstein-Sato polynomialof a (germ of an) analytic function will not be covered here.

For a field of characteristic zero K, let An(K) = K[x, ∂] be the Weyl algebra, that is, thenon-commutative ring of algebraic differential operators with x = x1, . . . , xn, ∂i = ∂/∂xi,∂ = ∂1, . . . , ∂n, and the usual relations ∂ixj − xj∂i = δij.

Let f ∈ K[x] be a non-constant polynomial. Let s be a dummy variable and K(s) thefield of rational functions in the variable s. Let

M = K(s)[x, f−1]f s,

be the free rank one K(s)[x, f−1]-module with the generator denoted f s. This is also anatural left An(K(s))-module, where the left An(K(s)) action is defined (as expected) by

∂j(gfs) =

(∂jg + sg∂j(f)f−1

)f s, xj(gf

s) = xjgfs,

for g ∈ K(s)[x, f−1].If we can show that M has finite length as a left An(K(s))-module, then one can construct

a non-zero polynomial b(s) and an operator P (s) as in (1.5.1). To see this, consider thedecreasing filtration of M by An(K(s))-submodules

An(K(s)) · f vf s,for v = 1, 2, . . . By the finite length assumption, there is w ∈ Z>0 such that R(s)fw+1f s =fwf s for some R(s) ∈ An(K(s)). Since s is a dummy variable, we can replace it with s+w,that is, we can assume w = 0. Let b(s) be a common denominator of the coefficients in R(s)of the monomials xα∂β. Then b(s) and P (s) = b(s)R(s) satisfy (1.5.1).

The fact that M has finite length as a left An(K(s))-module is a consequence of M beinga holonomic An(K(s))-module. We keep the notation simple and work from now with aleft An(K)-module M . To explain what holonomicity is, we first explain why M being afinitely generated An(K)-module is equivalent to M admitting a special kind of filtration.On An(K) there is the increasing Bernstein filtration F of K-vector spaces defined by

FpAn(K) = SpanK{xα∂β | |α|+ |β| ≤ p}.The associated graded vector space

GrFAn(K) = ⊕pFp/Fp−1is a graded commutative ring due to the fact that Fp · Fq ⊂ Fp+q. In fact, GrFAn(K) isisomorphic with the polynomial ring in 2n variables over K.

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A filtration F on M is a filtration of K-vector spaces such that ∪pFpM = M and FpAn(K)·FqM ⊂ Fp+qM . In this case, one has an associated graded GrFAn(K)-module GrFM , andwe say the F is a good filtration if GrF (M) is a finitely generated GrFAn(K)-module. Thefollowing is not too difficult to show:

Lemma 1.11. M is a finitely generated left An(K)-module iff M admits a good filtration.

Now we explain what it means for a finitely generated left An(K)-module M to be of finitelength. Since dimK FpM =

∑i≤p dimK Fi/Fi−1, the usual theory of graded modules finitely

generated over a polynomial ring gives that, for any good filtration F on M ,

dimK FpM = adpd + . . .+ a0

for some aj ∈ Q and p� 0. We let

d(M) = d, e(M) = d!ad.

Then d(M) and e(M) are two non-negative integers, called the dimension and the multiplicityof M . They are in fact independent of the choice of good filtration M , due to the following:

Lemma 1.12. Let F and F ′ be two filtrations on a left An(K)-module M . Assume F isgood. Then there exists q such that Fp ⊂ F ′p+q for all p.

Proof. Since F is good, there exists an integer p0 such that GrFM is generated over GrFAn(K)by elements of degree ≤ p0. Let q be such that Fp0M ⊂ F ′qM . One can show then thatFpM ⊂ F ′p+qM for all p. �

A fundamental result is Bernstein’s inequality:

Theorem 1.13. (Bernstein) If M is a non-zero finitely generated left An(K)-module, thend(M) ≥ n.

Definition 1.14. A non-zero finitely generated leftAn(K)-moduleM is holonomic if d(M) =n.

Proposition 1.15. Every strictly increasing sequence of An(K)-submodules of a holonomicmodule M contains at most e(M) terms. In particular, M has finite length.

Proof. If the sequence would be infinite, the multiplicities e would be ever increasing. Thisfollows from the property that under short exact sequences of An(K)-modules

(1.15.1) 0→M1 →M2 →M3 → 0,

d(M2) = max{d(M1), d(M3)} and e(M2) = e(M1) + e(M3). However, the multiplicity isbounded by e(M). �

There is a useful numerical criterion to guarantee that a module is holonomic:

Proposition 1.16. Let M be a left An(K)-module. If F is a filtration on M and if thereexist positive integers c1 and c2 such that dimK FpM ≤ c1p

n + c2(p+ 1)n−1 for all p, then Mis holonomic.

Proof. Let M0 be a finitely generated submodule of M . Let F ′ be a good filtration on M0.M0 has also the induced filtration F from M . By Lemma 1.12, we can find q such thatF ′pM0 ⊂ Fp+qM0 for all p. Then dimK F

′pM0 ≤ dimK Fp+qM ≤ c1(p+q)n+c2(p+q+1)n−1 ≤

c1pn + c3(p + 1)n−1, for a new constant c3. This implies that d(M0) ≤ n, hence d(M0) = n

5

and e(M0) ≤ n!c1. So, every finitely generated An(K)-submodule of M is holonomic andhas length ≤ n!c1 over An(K). It follows that the same holds for M . �

By our previous discussion, the following implies the existence of the non-zero Bernstein-Sato polynomial:

Proposition 1.17. Let f ∈ K[x] = K[x1, . . . , xn]. Let M be the left An(K(s))-moduleK(s)[x, f−1]f s. Then M is holonomic over An(K(s)).

Proof. Let

FpM = SpanK(s){gf−pf s | g ∈ K(s)[x], degx g ≤ (deg f + 1)p}.

Then dimK(s) FpM ≤ (deg f + 1)npn/n! + c2(p+ 1)n−1 for some constant c2. One also checksthat F forms a filtration. Then, by Proposition 1.16, M is holonomic over An(K(s)). �

1.18. Example: hyperplane arrangements. We summarize what is known about Bernstein-Sato polynomials for hyperplane arrangements. This highlights rather how little is known.

A polynomial f ∈ C[x1, . . . , xn] defines a hyperplane arrangement in Cn if it splits as aproduct of linear polynomials. An arrangement is reduced if f is. An arrangement is centralif f is homogeneous, and it is essential if it is not the pullback of an arrangement on a smalleraffine space. An arrangement f is indecomposable if it cannot be written as the product oftwo non-constant polynomials in two disjoint sets of variables, for any choice of coordinates.

Theorem 1.19. (M. Saito) The roots of the Bernstein-Sato polynomial bf (s) of a centralessential hyperplane arrangement f with d = deg f lie in the interval (−2 + 1/d, 0), and themultiplicity of the root s = −1 is n.

Theorem 1.20. (U. Walther) With same assumptions as above, if the hyperplanes formingf are in general position and d = deg f > n, then

bf (s) = (s+ 1)n−12d−2∏k=n

(s+

k

d

).

Conjecture 1.21. (Budur-Mustata-Teitler [BMZ-11]) Let f be an indecomposable essentialcentral hyperplane arrangement in Cn of degree d. Then bf (−n/d) = 0.

The importance of this conjecture lies in the fact that if true, it would prove the StrongMonodromy Conjecture for hyperplane arrangements. The Strong Monodromy Conjecture,which we will not state in these notes, ties the Bernstein-Sato polynomials with Igusa-Denef-Loeser zeta functions.

Theorem 1.22. Conjecture 1.21 holds for:

(a) (Budur-Saito-Yuzvinsky [BSY-11]) Reduced f with n ≤ 3, and for some other specialcases;

(b) (Bapat-Walters [BaW-15]) Finite Coxeter hyperplane arrangements;(c) (U. Walther [Wa-15]) Tame hyperplane arrangements. This case implies the previous

two cases.

Theorem 1.23. (U. Walther [Wa-15]) The Bernstein-Sato polynomial of a hyperplane ar-rangement is not a combinatorial invariant.

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An invariant is said to be combinatorial if it only depends on the lattice of intersection ofthe hyperplanes together with the codimensions of these intersections, and it does not dependon the position of the hyperplanes. Here is U. Walther’s example, worked out completely byM. Saito too:

Example 1.24. ([Wa-15]) Consider

f = xyz(x+ 3z)(x+ y + z)(x+ 2y + 3z)(2x+ y + z)(2x+ 3y + z)(2x+ 3y + 4z),

g = xyz(x+ 5z)(x+ y + z)(x+ 3y + 5z)(2x+ y + z)(2x+ 3y + z)(2x+ 3y + 4z).

These two hyperplane arrangements are combinatorially equivalent, but their Bernstein-Satopolynomials differ by one root:

bf (s) = (s+ 1)4∏i=2

(s+ i/3)16∏i=3

(s+ i/9),

bg(s) = (s+ 1)4∏i=2

(s+ i/3)15∏i=3

(s+ i/9).

1.25. The geometry behind classical Bernstein-Sato polynomials.

Definition 1.26. Let f : (Cn, 0) → (C, 0) be the germ of an analytic function such thatf(0) = 0. The Milnor fiber of f at 0 is

Ff,0 := f−1(t) ∩B,where B is a small ball around the origin and t is very close to 0.

A classical theorem of Milnor in the isolated singularity case, and Hamm-Le in general,state that the Milnor fiber Ff,0 is well-defined as a diffeomorphism class. The cohomologyvector spaces H i(Ff,0,C) admit an action called the monodromy generated by going oncearound a loop around 0 ∈ C. Another classical theorem due to Landmann, Grothendieck,etc., is:

Theorem 1.27. (Monodromy Theorem) Let f : (Cn, 0)→ (C, 0) be the germ of an ana-lytic function such that f(0) = 0. The eigenvalues of the monodromy action on H

q(Ff,0,C)

are roots of unity.

Example 1.28. (Milnor) When f has an isolated singularity,

dimCHj(Ff,0,C) =

0 for j 6= 0, n− 1,1 for j = 0,

dimC C[[x1, . . . , xn]]/(∂f∂x1, . . . , ∂f

∂xn

)for j = n− 1.

The last dimension is the Milnor number of f and can be computed by the computer programsmentioned earlier.

The following classical result of Malgrange and Kashiwara shows the geometric contentbehind the Bernstein-Sato polynomials.

Theorem 1.29. (Malgrange, Kashiwara) Let f : (Cn, 0)→ (C, 0) be the germ of an analyticfunction. Then:

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(a) The set Zero(bf ) of roots of the Bernstein-Sato polynomial of f consists of negativerational numbers.

(b) The setExp (Zero(bf )) = {exp(2πiα) | bf (α) = 0}

is equal to the set⋃x∈f−1(0)

x close to 0

⋃i∈Z

{eigenvalues of the monodromy on H i(Ff,x,C)}.

Remark 1.30. Note that this result also implies the Monodromy Theorem. In particu-lar, the Bernstein-Sato polynomials provide a way to use computers to calculate the set ofeigenvalues of the monodromy on the cohomology of Milnor fibers.

2. Bernstein-Sato polynomials of ideals and varieties

2.1. Bernstein-Sato polynomials for ideals. Let K be a field of characteristic zero andlet K[x] = K[x1, . . . , xn] the polynomial ring over K. Consider an ideal I ⊂ K[x] togetherwith a set of generators

I = 〈f1, . . . , fr〉 ⊂ K[x]

with fi ∈ K[x]. Then An(K) acts naturally on

K[x,r∏i=1

f−1i , s1, . . . , sr]r∏i=1

f sii .

Define an An(K)-linear action ti for i = 1, . . . , r by

ti(sj) =

{sj + 1 if i = jsj if i 6= j

This is applied to sj appearing as powers fsjj as well. For example, ti

∏j f

sjj = fi

∏j f

sjj .

Let sij = sit−1i tj for i, j ∈ {1, . . . , r}.

Definition 2.2. (Budur-Mustata-Saito [BMS-06a]) The Bernstein-Sato polynomial bI(s) ofthe ideal I = 〈f1, . . . , fr〉 ⊂ K[x] is defined to be the non-zero monic polynomial bI(s) ∈ K[s]of the lowest degree in s =

∑ri=1 si among those b(s) ∈ K[s] satisfying the relation

(2.2.1) b(s)r∏i=1

f sii =r∑j=1

Pjfj

r∏i=1

f sii ,

for some Pj ∈ An(K)[sij]1≤i,j,≤r. For h ∈ K[x], we define similarly

bI,h(s) ∈ K[s]

with∏

i fsii replaced by

∏i f

sii h.

Remark 2.3. One can prove that the definition of bI,h is independent of the choice ofgenerators for the ideal I. Note that bI,1(s) = bI(s). Also, b〈f〉,1(s) = bf (s) is the classicalBernstein-Sato polynomial defined when r = 1, cf. Definition 1.5.

Remark 2.4. We will see later that the existence of generalized Bernstein-Sato polynomialscan be reduced to the hypersurface case. In particular, the roots of bI(s) are all negativerational numbers as well.

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Remark 2.5. The first algorithm implemented for calculation of Bernstein-Sato polyomialsbI,h was due to T. Shibuta [Sh-11], building on the more precise description of bI,h from[BMS-06a] and on T. Oaku’s algorithm [Oa-97].

Example 2.6. Let I = 〈x2x3, x1x3, x1x2〉. Then bI(s) = (s+ 3/2)(s+ 2)2 and the operatorssij with i 6= j cannot be avoided by the operators Pj in the above definition.

Remark 2.7. More generally, the roots of the Bernstein-Sato polynomials bI(s) for monomialideals I have been determined combinatorially, in two ways, in [BMS-06b, BMS-06c]. It isinteresting to remark that [BMS-06c] uses positive characteristic techniques. However, themultiplicities of these roots were not addressed so far, and this remains an open question.

2.8. Bernstein-Sato polynomials of schemes. Let I = 〈f1, . . . , fr〉 ⊂ K[x] be an ideal.Let Z denote the closed subscheme of An

K defined by the ideal I. Let c be the codimensionof Z in An

K .

Theorem 2.9. ([BMS-06a]) The polynomial

bZ(s) := bI(s− c)depends only on Z and not on I.

Definition 2.10. We call bZ(s) the Bernstein-Sato polynomial of the scheme Z. For non-affine schemes Z, bZ(s) is defined as the lowest common multiplier of the Bernstein-Satopolynomials of the affine pieces; one can show this is well-defined.

2.11. Example: generic determinantal varieties. This is actually a challange, ratherthan an example. The idea is to generalize the formula from Cayley’s example for theBernstein-Sato polynomial of the determinant of a generic square matrix.

Let M = Mm,n = Cmn be the set of all complex m× n matrices. Consider the subset

Zk = Zk,m,n ⊂M

consisting of matrices with rank < k. Then Zk is the subvariety of M with ideal Ik generatedby the minors of size k of the matrix of indeterminates

(xij)1≤i≤m;1≤j≤n.

The basic question is to find formula for the Bernstein-Sato polynomial of Zk, or equiva-lently, of Ik.

Define

ck(s) =k−1∏i=0

(s+

(m− i)(n− i)k − i

).

The roots of the polynomial ck(s) are the poles of the Igusa-Denef-Loeser zeta functionof Ik, by work of Roi Docampo [Do-13]. The proof in [Do-13] is for the case m = n, buthe has informed us that the case m 6= n follows by a similar proof. Therefore the StrongMonodromy Conjecture implies that ck(s) divides the Bernstein-Sato polynomial bIk(s).

Around 2009, we stated a conjecture: if m = n, then ck(s) = bIk(s). When k = m = nthis states that the polynomial appearing in Proposition 1.2 is Bernstein-Sato polynomialof the determinant of the square matrix of indeterminates, which is well-known. The casek 6= n = m does not seem to fit into the setup of prehomogeneous vector spaces as inProposition 1.3. The conjecture was disproved around 2013 by a computer calculation:

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Example 2.12. (T. Oaku) Let m = n = 3, k = 2, so that I2 is the ideal of 2-by-2 minors ofthe matrix x11 x12 x13

x21 x22 x23x31 x32 x33

.

T. Oaku has computed that

bI2(s) = (s+ 9/2)(s+ 4)(s+ 5).

The factor s+ 5 does not appear in c2(s).

Example 2.13. Let k = 2,m = 2, n = 3, so that I2(s) is generated by the 2-minors in thematrix (

x11 x12 x13x21 x22 x23

).

In this case,

bI2(s) = (s+ 2)(s+ 3) = c2(s).

Example 2.14. Let k = 2,m = 2, n = 4, so that I2(s) is generated by the 2-minors in thematrix (

x11 x12 x13 x14x21 x22 x23 x24

).

In this case,

bI2(s) = (s+ 3)(s+ 4) = c2(s).

Example 2.15. (Raicu-Walther-Weyman) Let k = n ≤ m, so that In is the ideal generatedby the maximal minors of the generic m-by-n matrix of indeterminates. Then bIn(s) is adivisor of the polynomial

ck(s) = (s+m) . . . (s+m− n+ 1).

Example 2.16. (T. Shibuta) In the example above, the equality bIn(s) = (s + m) . . . (s +m− n+ 1) has been checked by T. Shibuta n = 2 and 2 < m < 13, and (n,m) = (3, 4).

For other examples related to matrices, see the article [CSS-13].The examples 2.12, 2.14, 2.16 are currently the most complex examples of Bernstein-Sato

polynomials for ideals of type Ik that computers can handle.

2.17. Relation with multiplier ideals. The standard reference for multiplier ideal is thetextbook [La-04]. Let X be a smooth complex algebraic variety or a complex manifold ofdimension n. Let Z be any closed subscheme or closed analytic subscheme of X. Chose localgenerators

IZ = 〈f1, . . . , fr〉of the ideal of Z.

Definition 2.18. The multiplier ideal of (X,αZ) for α ∈ R>0 is locally defined as

J (X,αZ) =

{g ∈ OX |

|g|2

(∑

i |fi|2)αis locally integrable

}.

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In fact, there is an equivalent geometric definition. Let µ : X ′ → X be a log resolutionof (X,Z), that is, X has been blown up sufficiently many times such that X ′ is smooth andthe ideal in OX′ generated by the image of the ideal of Z is the ideal of a hypersurface Hgiven locally by a monomial function. Let KX′/X denote the zero locus of the determinantof the Jacobian of µ.

Proposition 2.19. For α ∈ R>0,

J (X,αZ) = µ∗OX′(KX′/X − bαHc),

where b qc takes the round-down coefficient-wise for the irreducible components.

From this one sees that J (X,αZ) are coherent ideal sheaves and are independent of thechoice of generators for the ideal of Z, and from the definition one sees that the geometricinterpretation is independent of the choice of log resolution µ.

If

J (X,αZ) ( J (X, (α− ε)Z) for 0 < ε� 1,

we call α a jumping number of (X,Z). The smallest jumping number is called the logcanonical threshold and it is denoted lct (X,Z). Hence lct (X,Z) is the smallest α > 0 suchthat J (X,αZ) 6= OX .

Theorem 2.20. ([BMS-06a])

(a) For all α > 0, locally

J (X,αZ) = {h ∈ OX | α < c if bI,h(−c) = 0}.

(b) lct (X,Z) is the biggest root of bI(−s).(c) Any jumping number of (X,Z) in the interval [lct (X,Z), lct (X,Z) + 1) is a root of

bI(−s).

Remark 2.21. T. Shibuta [Sh-11] has used this theorem to implement the first algorithm,available for any closed subscheme of the affine space, for computations of multiplier ideals,log canonical thresholds, and jumping numbers of multiplier ideals.

So far we have not said much about the methods involved in the proofs of Theorems 1.29and 2.20, nor we have said anything about why non-zero Bernstein-Sato polynomials of typebI,h exist. The concept underlying these proofs is particular to the theory of D-modules:the theory of V -filtrations. We will introduce V -filtrations latter. This will also reveal thegeometry behind the Bernstein-Sato polynomials of ideals.

3. V -filtration

The underlying concept used in linking Bernstein-Sato polynomials with geometry isthe concept of V -filtrations. We introduce this concept in this section indirectly: via theRiemann-Hilbert correspondence. The reason is that this correspondence provides the ge-ometry behind the whole theory of D-modules. The V -filtration can be seen as a particularcase of explicit Riemann-Hilbert correspondence. In this section we will also sketch theproofs of Theorems 1.29, 2.20, along with the existence of bI,h(s), we will give the geometrybehind them, and we will finish with the mentioning that the multiplier-ideal filtrations arerestrictions of V -filtrations.

11

3.1. Riemann-Hilbert correspondence. A more flexible definition of cohomology in al-gebraic topology is through constructible sheaves. Recall that a constructible sheaf on acomplex analytic variety X is a sheaf of finite-dimensional C-vector spaces such that thereexists a stratification in the analytic topology of X with the property that the restriction ofthe sheaf to any element of the stratification is a locally constant sheaf (i.e. local system).Constructible sheaves form an abelian category, that is, roughly speaking, kernels and cok-ernels are again constructible sheaves. Thus one can form the bounded derived category ofconstructible sheaves Db

c(X) consisting of complexes of sheaves Fqwith constructible coho-

mology sheaves H i(Fq) which vanish for |i| � 0, and inverting quasi-isomorphisms. The

natural functors on constructible sheaves extend to derived functors on the derived cate-gory. For example, for a morphism p : X → Y , the direct image functor p∗ on constructiblesheaves, extends to the derived direct image functor Rp∗ such that, given a short exactsequence of complexes

0→ Fq

1 → Fq

2 → Fq

3 → 0,

one has a long exact sequence

. . .→ H i(Rp∗Fq

1 )→ H i(Rp∗Fq

2 )→ H i(Rp∗Fq

3 )→ H i+1(Rp∗Fq

1 )→ . . .

If X is a complex analytic manifold and f : X → C is a holomorphic function. Considerthe diagram

f−1(0) ��

i// X

f

��

X ×C C∗poo

��

C C∗? _oo C∗qoo

where C∗ is the universal cover of C∗, and p : X ×C C∗ → X is the natural projection. Onehas Deligne’s nearby cycles functor:

ψf := i∗Rp∗p∗ : Db

c(X)→ Dbc(f−1(0)).

Let CX be the constant sheaf on X. If x is a point such that f(x) = 0, ix : {x} → f−1(0)is the natural inclusion, and Ff,x is the Milnor fiber of f at x:

Theorem 3.2. (Deligne) Let X be a complex analytic manifold and f : X → C a holomor-phic function. Then

H i(Ff,x,C) = H i(i∗xψfCX)

and the action induced from the deck transformation of the covering C∗ → C∗ recovers themonodromy action on the cohomology of the Milnor fiber.

We now explain how the geometric picture of Milnor fibers can be understood via D-modules. Let X be nonsingular of dimension n. If X is algebraic, the sheaf of algebraiclinear differential operators DX is locally given in affine coordinates by the Weyl algebraAn(C). If X is analytic, one takes DX to be the sheaf of analytic linear differential op-erators. An important class of (left) DX-modules consists of those regular and holonomic.We have already introduced holonomicity, but we will not say more about the regularity.Let Db

rh(DX) be the bounded derived category of complexes of DX-modules with regularholonomic cohomology.

12

The topological package, consisting of the bounded derived category of constructiblesheaves Db

c(X) and the natural functors attached to it, has a D-theoretic counterpart. Thereis a well-defined functor, called the de Rham functor,

DRX : Dbrh(DX)→ Db

c(X),

that is an equivalence of categories commuting with the usual functors, and such that itrestrict to an equivalence between the abelian category of regular holonomic D-modulesand that of perverse sheaves on X. This is the famous Riemann-Hilbert correspondence ofKashiwara and Mebkhout.

Thus, there is a D-module counterpart of the nearby cycles functor ψf , hence of the Milnormonodromy for a function f : X → C. In concrete terms, this is achieved by the V -filtrationwhich we will introduce soon. However, let us state the result. Let

ψf = ⊕λψf,λ,with λ roots of unity, be the functor decomposition corresponding to the eigenspace decom-position of the semisimple part of the Milnor monodromy. Let

if : X → X × Cbe the graph embedding of f sending x to (x, f(x)).

Theorem 3.3. (Malgrange, Kashiwara) Let X be complex manifold of dimension n andf : X → C a holomorphic function. For α ∈ (0, 1],

ψf,λCX [n− 1] = DRX(GrαV (if )+OX),

where λ = exp(−2πiα) and (if )+ is the D-module direct image.

In the case of a regular function on a nonsingular algebraic variety, one should view thisas relating on one hand classical singularity theory, represented on the left-hand side bythe Milnor fiber monodromy via the theorem above due to Deligne, with, on the otherhand, purely algebraic data given by some construction with algebraic differential operators,represented on the right-hand side by the graded with respect to the V -filtration of someexplicit D-module.

We give more details about the V -filtration next.

3.4. V -filtrations on D-modules. Let X = Cn be the complex affine n-space. Let DX

be the Weyl algebra An(C). Let Y = X × Cr. Denote by OX = C[x], OY = C[x, t], withx = x1, . . . , xn, and t = t1, . . . , tr. So the ideal I ⊂ OY of the smooth closed subvarietyX × 0 of Y is generated by t. With this notation, DX = OX [∂x], DY = OY [∂x, ∂t], with∂x = ∂x1 , . . . ∂xn , ∂xi = ∂/∂xi, and similarly for ∂t. We will consider only left D-modules.

Definition 3.5. The filtration V along X × 0 on DY is

V jDY := { P ∈ DY | PI i ⊂ I i+j for all i ∈ Z },with j ∈ Z and I i = OY for i ≤ 0. So V jDY is generated over DX by the monomials tβ∂γtwith |β| − |γ| ≥ j.

Remark 3.6. By computation with local coordinates, one can show:(i) V j1DY · V j2DY ⊂ V j1+j2DY , with equality if j1, j2 ≥ 0;(ii) V jDY = Ij · V 0DY ·DY,−j = DY,−j · V 0DY · Ij, where DY,j ⊂ DY are the operators of

order ≤ j, and Ij = DY,j = OY for j ≤ 0.13

Let M be a finitely generated DY -module.

Definition 3.7. The filtration V along X × 0 on M is an exhaustive decreasing filtration offinitely generated V 0DY -submodules V α := V αM , such that:

(i) {V α}α is indexed left-continuously and discretely by rational numbers, i.e. V α =∩β<αV β, every interval contains only finitely many α with GrαV 6= 0, and these α must berational. Here, GrαV = V α/V >α, where V >α = ∪β>αV β.

(ii) tjVα ⊂ V α+1, and ∂tjV

α ⊂ V α−1 for all α ∈ Q, i.e. (V iDY )(V αM) ⊂ V α+iM ;(iii)

∑j tjV

α = V α+1 for α� 0;

(iv) the action of∑

j ∂tj tj − α on GrαV is nilpotent.All conditions depend only on the variety Y = X×Cr together with the closed subvariety

X × 0 and are independent of the choice of coordinates.

Theorem 3.8. (Malgrange, Kashiwara) The filtration V along X × 0 on M exists if M isregular holonomic and quasi-unipotent.

We have seen already what holonomic D-modules are. It suffices for our purposes to saythat all the D-modules considered in this section are regular holonomic and quasi-unipotent,without introducing these terms. We will show later how the proof of existence reduces tothe case r = 1.

Lemma 3.9. The filtration V along X × 0 on M is unique if it exists.

Proof. Say V is another filtration on M satisfying Definition 3.7. By symmetry it will be

enough to show that V α ⊂ V α for every α.Suppose that α 6= β and consider

V α ∩ V β/(V >α ∩ V β) + (V α ∩ V >β).

Since both filtrations satisfy 3.7-(iv), it follows that both (∑

j ∂tj tj − α) and (∑

j ∂tj tj − β)are nilpotent on this module. Hence the module is zero.

We show now that for every α we have

(3.9.1) V α ⊂ V >α + V α.

Fix m ∈ V α. By exhaustion, there is β � 0 (in particular β < α) such that m ∈ V β. By what

we have already proved, we may write m = m1 + m2, with m1 ∈ V >α and m2 ∈ V α ∩ V >β.

Say m2 ∈ V β1 with β1 > β. If we replace m by m2 and β by β1, then the class in V α/V >α

remains unchanged, and repeat the process. Since the filtration V is discrete, after finitelymany steps we have β ≥ α. Hence the class of m in V α/V >α can be represented by an

element in V α, and we get (3.9.1).Since the V -filtration is discrete, a repeated application of (3.9.1) shows that for every

β ≥ α we have V α ⊂ V β + V α. We deduce from 3.7-(iii) that if we fix β � 0, then

(3.9.2) V α ⊂ Iq · V β + V α

for q ∈ N. By coherence, V β =∑V 0DY · mi for finitely many mi. By exhaustion, there

exists some γ ∈ Z such that V γ contains the mi, hence also V β. By 3.7-(ii), for q with

q + γ ≥ α we have IqV γ ⊂ V α. Thus IqV β ⊂ V α. Hence by (3.9.2) we have V α ⊂ V α. �14

3.10. Bernstein-Sato polynomials of sections of D-modules. We keep the notation asabove.

Definition 3.11. Let M be a finitely generated DY -module. For m ∈ M , the Bernstein-Sato polynomial bm(s) of m is the non-zero monic minimal polynomial of the action ofs = −

∑j ∂tj tj on V 0DY ·m/V 1DY ·m.

Proposition 3.12. (Sabbah [Sa-84]) If the filtration V along X×0 on M exists, then bm(s)exists for all m ∈M , and has all roots rational. Moreover,

V αM = { m ∈M | α ≤ c if bm(−c) = 0 }.

Proof. Suppose that the filtration V exists on M . Let m ∈M . Then m ∈ V αM for some α,since V is an exhaustive filtration. Recall that

∑j ∂tj tj − β is nilpotent on V β/V >β and V

is indexed discretely. Then, for a given β there is a polynomial b(s) depending on β, havingall roots ≤ −α and rational, and such that b(−

∑j ∂tj tj) ·m ∈ V β. Hence it is enough to

show that there is β such that V β ∩ V 0DYm ⊂ V 1DYm.Let A =

⊕i≥0 V

iDY τ−i and define FkA =

⊕i≥0(V

iDY ∩DY,k)τ−i. Then F0A and GrFA

are noetherian rings and GrFkA are finitely generated F0A-modules for all k. It follows thatA is also noetherian.

Now⊕

i≥0 ViM is finitely generated over A because by axiom (iii) of 3.7, there exists

i0 such that V iM is recovered from V i0M if i ≥ i0. Denote by N the V 0DY -submoduleV 0DYm, and let U i = V i ∩ N for i ≥ 0. Then

⊕i≥0 U

iN is also finitely generated over A

since A is noetherian. It follows that⊕

i≥0 GriUN is finitely generated over⊕

i≥0 GriVDY . If

i is big compared with the degrees of local generators, we see that U iN ⊂ V 1DYm, which iswhat we wanted to show.

Conversely, fix an elementm ∈M and suppose that α ≤ c whenever bm(−c) = 0. Let αm =max{β | m ∈ V β}. We need to show that α ≤ αm. It is enough to show that bm(−αm) = 0.For β 6= αm, (

∑j ∂tj tj − β) is invertible on V αm/V >αm . But bm(−

∑j ∂tj tj)m ∈ V >αm .

Hence we must have bm(−αm) = 0. �

One can sheafify easily what we have discussed in this subsection and obtain the corre-sponding statements for the V -filtration on a coherent DY -module along a smooth subvarietyof Y of codimension r.

3.13. The geometry behind the V -filtration. Let X = Cn and f = (f1, . . . , fr) withfi ∈ C[x] = OX . Consider the graph embedding of f given by

if : X → X × Cr = Y, x 7→ (x, f1(x), . . . , fr(x)).

Let t = (t1, . . . , tr) be the coordinates of Cr. Let (if )+OX be the D-module direct image.By definition, this means that

(if )+OX = OX ⊗C C[∂t1 , . . . , ∂tr ]

with the left DY -action given as follows: for g, h ∈ OX , and ∂νt = ∂ν1t1 . . . ∂νrtr ,

g(h⊗ ∂νt ) = gh⊗ ∂νt , ∂xi(h⊗ ∂νt ) = ∂xih⊗ ∂νt −∑j

∂fj∂xi

h⊗ ∂tj∂νt ,

∂tj(h⊗ ∂νt ) = h⊗ ∂tj∂νt , tj(h⊗ ∂νt ) = fjh⊗ ∂νt − νjh⊗ (∂νt )j ,15

where (∂νt )j is obtained from ∂νt by replacing νj with νj − 1.Two facts need to be mentioned: OX is a regular holonomic DX-module, and the direct

image (if )+ of such a DX-module is a regular holonomic quasi-unipotent DY -module. Thus,by Theorem 3.8, the V -filtration on (if )+OX along X × 0 exists. For m = h⊗ 1 ∈ (if )+OXwith h ∈ OX , we have a polynomial bm(s) as in Definition 3.11 and the polynomials bm(s)determine the V -filtration on (if )+OX by Proposition 3.12.

Remark 3.14. When r = 1, we have thus clarified what is meant by the right-hand side theequality in Theorem 3.3 which says that the graded pieces with respect to the V -filtrationon (if )+OX admit a geometric meaning in terms of Milnor fibers. It is moreover true thatthe action of monodromy is reflected in the following fashion in D-module theoretic terms:s = −∂tt on V >0(if )+OX/V 1(if )+OX corresponds to the logarithm of the unipotent part Tuof the monodromy T on ψfCX under the Jordan decomposition T = TsTu.

Let us say now what geometric meaning is behind the V -filtration in the case r > 1. Withthe notation as in 3.4, let t = (t1, . . . tr) be the last coordinates on Y = X×Cr. There existsa specialization of Y to the normal cone NX×0Y of X × 0 in Y . More precisely, a diagramof natural maps

Y

Y × C∗

q;;

� �

j//

��

Y

ρ

OO

p

��

NX×0Y? _oo

��C∗ � � // C 0? _oo

where the two bottom squares are cartesian (i.e. fiber products) and the top triangle iscommuting. This diagram corresponds to the diagram of natural maps of algebras

C[x, t]

��uuC[x, t, u, u−1]

⊕i∈Z t

−iC[x, t]⊗ ui // //oo⊕

i≤0 t−i/t−i+1 ⊗ ui

C[u, u−1]

OO

C[u]

OO

oo // // C[u]/u

OO

where t−i is the ideal 〈t1, . . . , tr〉−i in OY = C[x, t] for i ≤ 0 and t−i = C[x, t] for i ≥ 0.Here the two bottom squares are cocartesian (i.e. tensor products) and the top triangle iscommuting.

The effect of this is that it reduces the setup of r regular functions f = (f1, . . . , fr) to thesetup of only one regular function p : Y → C.

For the smooth complete intersection case, that is, for t = (t1, . . . , tr), one can define ther > 1 analog of the Deligne nearby cycles functor, the so-called Verdier specialization functor

SpX×0|Y : Dbc(Y )→ Db

c(NX×0Y ), Fq 7→ ψp(Rj∗q

∗Fq).

For the arbitrary case f = (f1, . . . , fr) defining a closed subscheme Z ⊂ X, one defines

SpZ|X : Dbc(X)→ Db

c(NZX), Fq 7→ SpX×0|Y (if )∗F

q.

16

In fact, for r = 1 this recovers the Deligne nearby cycles functor.Taking into account the monodromy from the map p : Y → C, there is a monodromy

action on SpZ|X with eigenvalues roots of unity, and a decomposition SpZ|X =⊕

λ SpZ|X,λgeneralizing the one for Deligne nearby cycles functor.

By Theorem 3.3, the D-module counterpart of SpZ|X,λCX is given by GrαV M where α ∈(0, 1], λ = exp(−2πiα), M = (if )+OX = C[x] ⊗ C[∂t] with the natural action of DY ,

M = j+q+M = ⊕i∈ZM ⊗ ui with the natural action of DY ⊗ C[u, ∂u], and V

qM is the

V -filtration along NX×0Y in Y . This V -filtration exists because M is regular holonomic andquasi-unipotent with respect to the map p.

Consider now the V -filtration on M along X × 0 on Y . With a bit of computation withcoordinates, one can show that it is related to the V -filtration on M :

V α−r+1M =⊕i∈Z

V α−iM ⊗ ui.

In fact, this defines the V -filtration on M along X × 0 in Y and reduces the proof of itsexistence to the r = 1 case. This also translates into a relation between Bernstein-Satopolynomials of m ∈M and m = m⊗ 1 in M with respect to the two V -filtrations:

(3.14.1) bm(s) = bm(s+ r − 1).

We summarize the discussion and draw the following geometric interpretation of the V -filtration which follows from the r = 1 case.

Theorem 3.15. ([BMS-06a]) Let X be a nonsingular complex variety.(a) Let f = (f1, . . . , fr) define a closed subscheme Z in X with fi : X → C regular

functions. For α ∈ (0, 1],

SpZ|X,λCX = DRNXY

(⊕i∈Z

Grα−iV (if )+OX ⊗ ui)

up to a shift, where λ = exp(−2πiα) and V is the filtration along X × 0 on (if )+OX .(b) Let m ∈ (if )+OX . The set Zero(bm) of roots of the Bernstein-Sato polynomial of m

consists of negative rational numbers.(c) The set Exp (Zero(bm)) is included in the set of eigenvalues of SpZ|XCX , with equality

if m generates (if )+OX .

This implies and generalizes Theorem 1.29.This theorem provides the existence, rationality, and the geometry behind the more general

Bernstein-Sato polynomials bI,h(s) for ideals via the following lemma. Let

I = 〈f1, . . . , fr〉.

Lemma 3.16. Let m = h⊗ 1 ∈ (if )+OX = OX [∂t]. Then

bI,h(s) = bm(s).

Proof. It is enough to show that bm(s) is the minimal polynomial of the action of s =∑

j sjon

DX [sij]∏j

fsjj h/

∑k

DX [sij]fk∏j

fsjj h,

17

a quotient of submodules of OX [∏

i f−1i , s1, . . . , sr]

∏i f

sii h. We can check that this quotient

is isomorphic to V 0DYm/V 1DYm. The action of tj agrees on both sides,∏

j fsjj h corresponds

to m, and sj corresponds to −∂tj tj. �

Remark 3.17. Indeed, by this lemma and by (3.14.1), the existence of bI(s) is reduced tothat of the classical Bernstein-Sato polynomials, that is, to the case r = 1, which we haveproved. Similarly for bI,h(s). It would be interesting to give a proof of the existence of bI,h(s)along the lines for the classical case without passing through the argument via specializationto the normal cone.

Theorem 3.15 gives the geometric interpretation of bI,h(s), generalizing Theorem 1.29:

Corollary 3.18. ([BMS-06a]) The exponentials of the roots of bI,h(s) are among the mon-odromy eigenvalues of SpZ|XCX . The exponentials of the roots of bI(s) give all the eigenval-ues.

Theorem 3.15 also implies, via the previous lemma, Theorem 2.20 since we have thefollowing D-module theoretic interpretation of multiplier ideals:

Theorem 3.19. ([BMS-06a]) The multiplier ideal filtration is the restriction of the V -filtration. More precisely, let X,Z, and f be as above. Then for all α > 0 and 0 < ε� 1,

J (X, (α− ε)Z) = (OX ⊗ 1) ∩ V α(if )+OX .

Remark 3.20. One should interpret this result, in case Z is a hypersurface given by afunction f , as relating the multiplier ideals with classical singularity theory, representedon the right-hand side by objects computing Milnor fiber monodromy. See [Bu-03, BS-05]for more details. A more precise description of the right-hand side requires the formalismof Hodge filtrations and the geometry behind the right-hand side is then expressed via M.Saito’s theory of mixed Hodge modules.

4. Bernstein-Sato ideals

Most of this section covers material from [Bu-15].

4.1. Bernstein-Sato ideals for many functions. Let X = Cn. Let F = (f1, . . . , fr) bea collection of polynomials fj in C[x1, . . . , xn]. Let D = ∪ri=1f

−1i (0). We will often identify

F with the associated mapping F : X → Cr, as opposed to what we did in the previoussections when we looked at the ideal generated by the fi. Let DX = An(C).

Definition 4.2. The Bernstein-Sato ideal of F = (f1, . . . , fr) with fi ∈ C[x1, . . . , xn] is theideal BF generated by polynomials b ∈ C[s1, . . . , sr] such that

b(s1, . . . , sr)fs11 · · · f srr = Pf s1+1

1 · · · f sr+1r

for some algebraic differential operator P ∈ DX [s1, . . . , sr].

The existence of non-zero Bernstein-Sato ideals BF is similar to that of the case r = 1 inDefinition 3.11 as shown by Lichtin [Li-88]. One can similarly define the local Bernstein-Satoideal BF,x of a finite collection of germs of holomorphic functions at a point in x ∈ Cn byusing analytic differential operators. The proof of the existence in this case is due to Sabbah[Sa-87], see also [Ba-05].

18

In the one-variable case r = 1, the monic generator of the ideal BF is the classicalBernstein-Sato polynomial. In general, the ideal BF is not always principal. The ideal BF isgenerated by polynomials with coefficients in the subfield of C generated by the coefficientsof F .

Example 4.3. If fj are monomials, write fj =∏n

i=1 xai,ji . Let li(s1, . . . , sr) =

∑rj=1 ai,jsj.

Let ai =∑r

j=1 ai,j. Then

BF = 〈n∏i=1

(li(s) + 1) · · · (li(s) + ai)〉.

The following would generalize Theorem 1.29:

Conjecture 4.4. ([Bu-15]) Let F = (f1, . . . , fr) with fi ∈ C[x1, . . . , xn].(a) The Bernstein-Sato ideal BF is generated by products of linear polynomials of the form

α1s1 + . . .+ αrsr + α

with αj ∈ Q≥0 and α ∈ Q>0.(b)

Exp (Zero(BF )) =⋃y∈D

Supp y(ψFCX).

Here, we let Exp : (C∗)r → Cr be the map α 7→ exp(2πα), and we let Zero(I) denote thezero locus of an ideal I.

The complex ψFCX in Dbc(X), which we call the Sabbah specialization complex, is defined

exactly like Deligne nearby cycles complex except now one replaces C∗ with (C∗)r. This isnow a complex of A-modules, where A = C[t±11 , . . . , t±r ] is the affine coordinate ring of thetorus S∗ = (C∗)r, and the tj denote monodromies with respect to the fj. Now, Supp x(ψFCX)denotes the union of all the supports in S∗ of the cohomology A-modules H i(ψFCX)x of thestalk at x.

In the case r = 1, this geometric picture recovers the monodromy eigenvalues of Milnorfibers, that is, Conjecture 4.4 implies Corollary 1.29.

Part (a) would refine a result of Sabbah [Sa-90] and Gyoja [Gy-93] which states that BF,x

contains at least one element of this type. Part (b) needs some definitions and an explanationas to why it would recover the Malgrange-Kashiwara property for r = 1 case.

Let x ∈ X. It is known thatBF =

⋂x∈D

BF,x.

Thus,

Zero(BF ) =⋃x∈D

Zero(BF,x).

Moreover, this is a finite union since there is a constructible stratification of X such thatfor x running over a given stratum the Bernstein-Sato ideal at x is constant. Thus we canand do make a local version of the Conjecture at the point x, from which the stated versionfollows

Let us say what has been proven about the Conjecture.

Theorem 4.5. With notation as above, we have:19

(a) ([Bu-15])

Exp (Zero(BF,x)) ⊃⋃

y∈D near x

Supp y(ψFCX),

and equality is conjectured. 1

(b) (Budur-Wang [BW-15b]) Supp y(ψFCX) is a finite union of torsion translated subtori of(C∗)r.

We will give a proof of Part (a) later. One can show that the remaining unproved portionof the Conjecture would follow from:

Conjecture 4.6. Let x ∈ X. Assume that F = (f1, . . . , fr) is such that the fj with fj(x) = 0define mutually distinct reduced and irreducible hypersurface germs at x. Then, locally at x,for all α ∈ Zero(BF,x),

r∑j=1

(sj − αj)DX [s1, . . . , sr]fs11 . . . f srr 6≡ DX [s1, . . . , sr]f

s11 . . . f srr

modulo DX [s1, . . . , sr]fs1+11 . . . f sr+1

r .

Example 4.7. Let F = (x, y, x+ y, z, x+ y+ z). Then the product of all entries of F formsa central essential indecomposable hyperplane arrangement in C3, the cone over these lines:

This means that we can easily compute the right-hand side of Conjecture 4.4, which thenpredicts that, in (C∗)5,

(4.7.1) Exp (Zero(BF )) = Zero(〈(t1t2t3 − 1)(t3t4t5 − 1)(t1 . . . t5 − 1)5∏j=1

(tj − 1)〉).

We will check this later, see Example 4.12. However, for now note that the Bernstein-Satoideal BF of F is currently intractable via computer.

4.8. Ideals of Bernstein-Sato type. There are many ways to define ideals of Bernstein-Sato type different than the one in Definition 4.2. They do help understanding BF however.

Let F = (f1, . . . , fr) with fj ∈ C[x1, . . . , xn]. Let

M = {mk ∈ Nr | k = 1, . . . , p}be a collection of vectors, which we also view as an p×r matrix M = (mkj) with mkj = (mk)j.

1As pointed out by Liu-Maxim [LM-14], in all the statements in [Bu-15] where the uniform supportSuppunifψFCX of the Sabbah specialization complex appears, the unif should be dropped to conform towhat is proven in [Bu-15].

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Definition 4.9. The Bernstein-Sato ideal associated to F and M is the ideal

BMF = B

m1,...,mp

F ⊂ C[s1, . . . , sr]

of all polynomials b(s1, . . . , sr) such that

b(s1, . . . , sr)r∏j=1

fsjj =

p∑k=1

Pk

r∏j=1

fsj+mkj

j

for some algebraic differential operators Pk in DX [s1, . . . , sr].

Remark 4.10. (a) BF , as defined before, is B 1F , where 1 = (1, . . . , 1).

(b) For a point x in X, the local Bernstein-Sato ideal BMF,x is similarly defined by replacing

DX with the ring DX,x of germs of holomorphic differential operators at x. Then

BMF =

⋂x∈X

BMF,x.

(c) The ideals BMF,x are non-zero by Sabbah.

Theorem 4.11. ([Bu-15]) Let m ∈ Nr. For j = 1, . . . , r, let tj be the ring isomorphism ofC[s1, . . . , sr] defined by tj(si) = si + δij. Then there are inclusions of ideals in C[s1, . . . , sr]∏

1≤j≤rmj>0

mj−1∏k=0

tm11 . . . t

mj−1

j−1 tkj · BejF ⊂ Bm

F ⊂⋂

1≤j≤rmj>0

mj−1⋂k=0

tm11 . . . t

mj−1

j−1 tkj · BejF .

Here δij = 0 if i 6= j, and δii = 1. Also, we denote by ej the r-tuple with the k-th entryδjk. By convention, t0j is the identity map, the product map ta11 . . . tarr means the obviouscomposition of maps, and ta11 . . . tarr I is the image of the ideal I under this product map.

Example 4.12. This result is useful in practice. Let us retake Example 4.7. Recall thatBF was intractable via computers. However, one can compute with dmod.lib [LM-11]:

Be1F = 〈(s1 + 1)(s1 + s2 + s3 + 2)(s1 + s2 + s3 + s4 + s5 + 3)〉,

Be2F = 〈(s2 + 1)(s1 + s2 + s3 + 2)(s1 + s2 + s3 + s4 + s5 + 3)〉,

Be3F = 〈(s3 + 1)(s1 + s2 + s3 + 2)(s3 + s4 + s5 + 2)(s1 + s2 + s3 + s4 + s5 + 3)〉,

Be4F = 〈(s4 + 1)(s3 + s4 + s5 + 2)(s1 + s2 + s3 + s4 + s5 + 3)〉,

Be5F = 〈(s5 + 1)(s3 + s4 + s5 + 2)(s1 + s2 + s3 + s4 + s5 + 3)〉.

Then (4.7.1) follows from Theorem 4.11 which implies in particular that

Exp (V (BF )) =5⋃j=1

Exp (V (BejF )).

Theorem 4.11 is a consequence of the following result, in which we will use the notationtm =

∏rj=1 t

mj

j .

Lemma 4.13. Let m,n ∈ Nr. Then there are inclusions of ideals

BmF · (tmBn

F ) ⊂ Bm+nF ⊂ Bm

F ∩ (tmBnF ).

21

Proof. We will use the notation f s =∏r

j=1 fsjj . Let b1 ∈ Bm

F and b2 ∈ BnF . Write b1f

s =

P1fs+m and b2f

s = P2fs+n for some P1 and P2 in DX [s1, . . . , sr]. Apply tm to both sides of

b2fs = P2f

s+n. We obtain then that (tmb2)fs+m = (tmP2)f

s+m+n. Applying P1 on the lefton both sides of the equality, we have

P1(tmP2)f

s+m+n = P1(tmb2)f

s+m = (tmb2)P1fs+m = (tmb2)b1f

s.

Thus b1(tmb2) is in Bm+n

F , which implies the first inclusion.Take now b ∈ Bm+n

F . Write bf s = P f s+m+n for some P in DX [s1, . . . , sr]. Then bf s =P fnf s+m, so b ∈ Bm

F . Now, multiply by fm on the left on both sides of the last equality.We obtain that bf s+m = fmP f s+m+n. This shows that b ∈ tmBn

F . Hence b ∈ BmF ∩ (tmBn

F ),which proves the second inclusion. �

4.14. Relation with local systems. There is another, more concrete, description of thesupport loci Supp y(ψFCX) of the Sabbah specialization complex in terms of rank one localsystems.

For a topological space U , the spaceMB(U) of local systems of rank one on U is a groupunder the tensor product. It is identified by mondromies around loops with

MB(U) = Hom(H1(U,Z),C∗) = (C∗)r × (finite abelian group).

Define the cohomology jump loci of U

Σik(U) := {L ∈MB(U) | dimH i(U,L) ≥ k},

and the cohomology support loci to be

Σ(U) := {L ∈MB(U) | H q(U,L) 6= 0} =

⋃i

Σi1(U).

Coming back to our situation, for a point x in X, let UF,x be the complement of D in asmall open ball centered at x,

UF,x := Ballx − (Ballx ∩D).

There is a natural pullback map

F ∗ :MB((C∗)r) = (C∗)r →MB(UF,x)

induced by F . This is an isomorphism if the polynomials fj define the mutually distinctreduced and irreducible analytic branches of f =

∏j fj at x.

Theorem 4.15.Supp x(ψFCX) = (F ∗)−1(Σ(UF,x)).

The Part (b) of Theorem 4.5 is a particular case of the recent more difficult result:

Theorem 4.16. (Budur-Wang [BW-15b]) Σik(UF,x) is finite union of torsion translated

subtori in MB(UF,x).

By subtori, here it is mean the affine algebraic subtori of type (C∗)p for some p.This generalizes the Mondromy Theorem (Theorem 1.27) as well. The proof is however

non-constructive, in the sense that there is one burning question left unanswered:

Question 4.17. Is there a refined Bernstein-Sato type ideal Bik ⊂ C[s1, . . . , sr], depending

on F and x, such thatExp (Zero(Bi

k)) = Σik(UF,x) ?

22

4.18. Proof of Part (a) of Theorem 4.5. One can show with a bit of work that it isenough to restrict to the case when the fj with fj(x) = 0 define mutually distinct reducedand irreducible hypersurface germs at x. Let us assume this is the case from now. Hence,by Theorem 4.15, we have to prove a statement about rank one local systems. Namely, it isenough to show that

Exp (Zero(BF,x)) ⊃ Σ(UF,x).

Let α ∈ Cr such that λ = Exp (α) is not in Exp (Zero(BF,x)). Viewing λ ∈ (C∗)r asthe corresponding rank one local system Lλ ∈ MB(UF,x) = (C∗)r, with monodromy givenby multiplication by λi around the divisor fi, it is then enough to show that Lλ is not inΣ(UF,x). In other words, it is enough to show that Lλ has trivial cohomology on UF,x.

We will denote UF,x by Ux from now. Note that Lλ is actually the restriction to Ux of alocal system on U = X \D. We will sometimes abuse the notation and denote the latter byLλ also. The cohomology on U of Lλ can be different than the cohomology on Ux, so let uskeep this in mind.

Since α +m · 1 is not in Zero(BF,x) for any integer m,

Pα = Pα+m·1on Ux for all integers m, where

(4.18.1) Pα = DXr∏j=1

fαj

j .

Here DX is the sheaf of analytic differential operators on X, and Pα is a DX-submodule ofthe free rank one OX [f−1]-module with generator denoted

∏rj=1 f

αj

j and with the expected

derivation rules. The equation (4.18) follows from the fact that the Bernstein-Sato ideal BF,x

is non-zero.A piece of explicit Riemann-Hilbert correspondence states that,

Rj∗(Lλ[n]) = DRX(Pα+m·1) (m� 0),

and

ICX(Lλ) = DRX(Pα−m·1) (m� 0).

Thus, locally at x, we have

(4.18.2) Rj∗(Lλ[n]) = ICX(Lλ).

Here j : U → X is the open embedding of the complement U of D into X. Also, ICX(L)of a local system L is the intersection complex, also called the intermediate extension of theperverse shifted complex L[n], and defined in the abelian category of perverse sheaves on Xby

ICX(L) = j!∗(L[n]) = im{j!(L[n])→ Rj∗(L[n])}.We will prove that (4.18.2) implies that Lλ has trivial cohomology on Ux. At this point,

there is a mistake in the proof of [Bu-15], which goes on to say that it is automatically impliedthat Rj∗Lλ = j∗Lα locally at x. While this would indeed lead to the same conclusion, it isnot necessarily automatically true, unless D is a divisor with normal crossings at x.

Let L be any local system on U . There are short exact sequences in Perv(X),

(4.18.3) 0→ K(L)→ j!(L[n])→ ICX(L)→ 023

and

(4.18.4) 0→ ICX(L)→ Rj∗(L[n])→ C(L)→ 0.

The Verdier dual of the first sequence is the second sequence applied to the dual local systemL∨, and the other way around.

The interesting thing is that K(L) and C(L) are simultaneously zero or not. More gen-erally, they have the same lengths as perverse sheaves. This follows from the descriptionof K(L) and C(L) in terms of Deligne’s nearby cycles functor, namely, that there exists anexact sequence in Perv(X)

0→ K(L)→ pψf (L[n])TI−→ pψf (L[n])→ C(L)→ 0.

On the other hand, one has the distinguished triangle in Dbc(X),

j!(L[n])→ Rj∗(L[n])→ i∗i−1Rj∗(L[n])

+1−→obtained by applying the usual sequence

j!j−1 → id→ i∗i

−1 +1−→to Rj∗(L[n]), where i : D → X is the closed embedding. Taking perverse cohomology, onehas in Perv(X) an exact sequence

(4.18.5) 0→ pH−1(i∗i−1Rj∗(L[n]))→ j!(L[n])→ Rj∗(L[n])→ pH0(i∗i−1Rj∗(L[n]))→ 0,

and the vanishingpHj(i∗i

−1Rj∗(L[n])) = 0 for j 6= −1, 0.

Hence, there is a third description of K(L) and C(L):

K(L) = pH−1(i∗i−1Rj∗(L[n])),

C(L) = pH0(i∗i−1Rj∗(L[n])).

If one has, as we have for L = Lλ, that

ICX(L) = Rj∗(L[n])

locally at x ∈ D, then this implies the simultaneous vanishing of K(L) and C(L) at x, whichin turn implies that

i−1x (Rj∗(L[n])) = i−1x (i∗i−1Rj∗(L[n]))

is quasi-isomorphic with zero, where ix : {x} → X is the embedding of the point x. Notethat i−1x Rj∗(L) is the stalk of Rj∗(L) at x. Hence H i(i−1x Rj∗(L)) is the stalk of Rij∗(L). Thisis the same as the global sections of Rij∗(L) in a small ball Bx around x. By the definitionof the higher direct image functors on sheaves,

Γ(Bx, Rij∗(L)) = H i(Ux, L).

It follows that Lλ cannot have any cohomology on Ux, as claimed. �

Note that the simultaneous of appearance of a rank one local system Lλ and of its dual,or inverse, L∨ = Lλ−1 in the cohomology support locus used in the above proof is also aparticular case of a more general fact:

Theorem 4.19. Let X be a topological space such that Σik(X) is a finite union of torsion

translated subtori of MB(X). Then λ ∈ Σik(X) if and only if λ−1 ∈ Σi

k(X).

24

Proof. Let λ ∈ Σik(X). We can assume λ sits in a component ρ · T of Σi

k(X), where Tis a subtorus in MB(X) and ρ is torsion. Since Σi

k(X) is a variety defined over Q, it isinvariant under the action of the Galois group of Q over Q. Let σ be the action such thatσ(ρ) = ρ−1. Then σ(ρT ) must be a subset of Σi

k(X). But σ(ρT ) = ρ−1σ(T ) = ρ−1T , sinceany algebraic subtorus of (C∗)r is also defined over Q. If λ = ρ · t, for some t ∈ T , thenλ−1 = ρ−1 · t−1 ∈ ρ−1T = σ(ρT ) ⊂ Σi

k(X). �

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KU Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, BelgiumE-mail address: Nero.Budur@wis.kuleuven.be

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