BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS (LECTURE NOTES, - - PDF document

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS (LECTURE NOTES, ROLDUC ABBEY, 2013)

NERO BUDUR

• Abstract. These are lecture notes from a series of lectures at the Summer

school Algebra, Algorithms, and Algebraic Analysis, Rolduc Abbey, Netherlands, September 2-6, 2013.

Contents 1 1. Classical Bernstein-Sato polynomials 2 1.1. Original motivation. 2 1.7. Proof of existence. 4 1.15. Challenge: hyperplane arrangements 6 1.17. The geometry behind: Milnor fibers. 6 2. V -filtration 9 2.1. V -filtrations on D-modules. 9 2.9. The geometry behind the V -filtration. 11 3. Bernstein-Sato polynomials of varieties 13 3.1. Bernstein-Sato polynomials of varieties 13 3.6. Another relation with geometry: multiplier ideals. 15 3.11. Challenge: generic determinantal varieties. 15 4. Bernstein-Sato ideals for mappings 16 4.1. Bernstein-Sato ideals for mappings 16 4.8. Ideals of Bernstein-Sato type. 18 4.14. Cohomology support loci of local systems. 20 References 20 These are lecture notes from a series of lectures at the Summer school Algebra, Algorithms, and Algebraic Analysis, Rolduc Abbey, Netherlands, September 2-6,

• 2013. While the general purpose of the Summer school was on algebraic compu-

tations, the purpose of these lectures is to partially answer, in a manner related to the topic of Bernstein-Sato polynomials, the following: what are the algebraic

Date: August 28, 2013. This work was partially supported by the Simons Foundation grant 245850.

1

2 NERO BUDUR

algorithms computing geometrically, how can we improve or come up with new algorithms for other geometric invariants, and how can we use computations to predict previously-unknown behavior of geometric invariants. We can only cover a few topics here, and the choice is biased and reflects personal taste. These notes lack any guide to literature; for that, please see the extensive survey [B2].

• 1. Classical Bernstein-Sato polynomials

1.1. Original motivation. 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

• f indeterminates. Then

(s + 1)(s + 2) . . . (s + n)f s = det(∂/∂xij)f s+1. (1) Generalizations by M. Sato lead to the theory of prehomogeneous vector spaces. While the main goal is classification, functional relations such as (1) are of funda- mental importance. A prehomogeneous vector space is a vector space V over a field K of characteristic zero with an algebraic action ρ : G → GL(V ) of an algebraic group G such that it 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 com- plement V \U are given by homogeneous irreducible polynomials which are semi-

• invariants. Moreover, all semi-invariants are of this type. When K = C and G is

a complex reductive group, the dual action ρ∗ : g → 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 prehomo- geneous vector space with G. If f is a semi-invariant of degree d, there exists a non-zero polynomial b(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 matrices via the usual multiplication ρ(g) : x → gx. Then (G, V, ρ) is a prehomoge- neous vector space, f(x) = det(x) is a semi-invariant for the character χ : G → C∗ given by χ(g) = det(g), and Proposition 1.3 generalizes Proposition 1.2. Definition 1.5. Let f ∈ K[x1, . . . , xn] be a polynomial with coefficients in a field K of characteristic zero. The Bernstein-Sato polynomial of f is the non-zero monic polynomial bf(s) of minimal degree among those b ∈ K[s] such that b(s)f s = Pf s+1 (2) for some operator P ∈ K[x, ∂/∂x, s].

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS 3

A few things should be said here. Firstly, it is non-trivial that non-zero Bernstein- Sato polynomials exist. The existence was proved by I.N. Bernstein, independently

• f Sato’s proof for semi-invariants of prehomogeneous vector spaces. Secondly, by

a result of B. Malgrange and M. Kashiwara, the roots of bf(s) are in Q<0. We will come back to this later. Bernstein’s work was motivated by a question I.M. Gelfand posed in 1963: what is the meaning 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, if f(x) ≤ 0. Then the question is if f s

+ admits a meromorphic continuation to all s ∈ C and, if

so, to describe the poles. This was positively answered by M. Atiyah and Bernstein- Gelfand who described the poles in terms of a resolution of singularities of f. A more precise result was proved by Bernstein: Proposition 1.6. As a distribution, f s

+ admits a meromorphic continuation with

poles in the 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

functions φ, 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 β aβ(x, s). 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). This continues meromorphically the left-hand side to {s | Re(s) > −1} with poles in the zero locus of b(s). By iterating this process, we obtain the Proposition.

4 NERO BUDUR

1.7. Proof of existence. We now sketch the proof of the existence of a non-zero polynomial as in Definition 1.5. This will be a crash course on the basic theory of D-modules. For more details see [Bj]. For a field of characteristic zero K, let An(K) = K[x, ∂] be the Weyl algebra, that is, the non-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) the field of rational function in the variable s. Let M be the left An(K(s))- module generated by f s. That is M is the free rank one K(s)[x, f −1]-module with the generator denoted f s, M = K(s)[x, f −1]f s, and the left An(K(s)) action on M is defined by ∂j(gf s) =

• ∂jg + sg∂j(f)f −1

f s, xj(gf s) = xjgf s, 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 (2). To see this, consider the decreasing filtration 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)f w+1f s = f wf 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 (2). The fact that M has finite length as a left An(K(s))-module is a consequence of M being a holonomic An(K(s))-module. We keep the notation simple and work from now with a left An(K)-module M. To explain what holonomicity is, we first explain why M being a finitely 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−1 is a graded commutative ring due to the fact that Fp·Fq ⊂ Fp+q. In fact, GrFAn(K) is isomorphic with the polynomial ring in 2n variables over K. 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, and we say the F is a good filtration if GrF(M) is a finitely generated. The following is not too difficult to show: Lemma 1.8. M is a finitely generated left An(K)-module iff M admits a good filtration.

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS 5

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

i≤p dimK F i/F i−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 multiplicity of M. They are in fact independent of the choice of good filtration M, due to the following: Lemma 1.9. Let F and F ′ be two filtrations on a left An(K)-module M. Assume F is good. 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 that FpM ⊂ F ′

p+qM for all p.

• A fundamental result is Bernstein’s inequality:

Theorem 1.10. If M is a non-zero finitely generated left An(K)-module, then d(M) ≥ n. Definition 1.11. A non-zero finitely generated left An(K)-module M is holonomic if d(M) = n. Proposition 1.12. Every strictly increasing sequence of An(K)-submodules of a holonomic module 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 increas-
• ing. This follows from the property that under short exact sequences of An(K)-

modules (3) 0 → M1 → M2 → M3 → 0, d(M2) = max{d(M1), d(M3)} and e(M2) = e(M1) + e(M3). However, the multi- plicity is bounded by e(M).

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

Proposition 1.13. Let M be a left An(K)-module. If F is a filtration on M and if there exist positive integers c1 and c2 such that dimK FpM ≤ c1pn + c2(p + 1)n−1 for all p, then M is holonomic.

• Proof. Let M0 be a finitely generated submodule of M. Let F ′ be a good filtration
• n M0. M0 has also the induced filtration F from M. By Lemma 1.9, we can

find q such that F ′

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 and e(M0) ≤ n!c1. So, every finitely

6 NERO BUDUR

generated An(K)-submodule of M is holonomic and has 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 1.14. Let f ∈ K[x] = K[x1, . . . , xn]. Let M be the left An(K(s))- module K(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 checks that F forms a filtration. Then, by Proposition 1.13, M is holonomic

• ver An(K(s)).
• 1.15. Challenge: 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 a product of linear polynomials. An arrangement is reduced if f is. An arrangement is central if f is homogeneous, and it is essential if it is not the pullback

• f an arrangement on a smaller affine space. An arrangement f is indecomposable if

it cannot be written as the product of two non-constant polynomials in two disjoint sets of variables, for any choice of coordinates.

• M. Saito showed that the roots of a the Bernstein-Sato polynomial bf(s) of a

central essential hyperplane arrangement f are between (−2, 0) and the multiplicity

• f the root s = −1 is n.
• U. Walther showed that if the hyperplanes forming f are in general position and

d = deg f > n, then bf(s) = (s + 1)n−1

2d−2

• k=n
• s + k

d

• .

The following was conjectured by Budur-Mustat ¸˘ a-Teitler: Conjecture 1.16. Let f be an indecomposable essential central hyperplane ar- rangement in Cn of degree d. Then bf(−n/d) = 0. This is proved for reduced f with n ≤ 3, and for some other cases, by Budur- Saito-Yuzvinsky. Its importance lies in the fact that if true, this conjecture would prove the Strong Monodromy Conjecture for hyperplane arrangements, tying the Bernstein-Sato polynomials with Denef-Loeser zeta functions. 1.17. The geometry behind: Milnor fibers. Although the definition of the classical Bernstein-Sato polynomial is elementary, to understand the underlying geometrical meaning is not. We recall now how, via V -filtrations on D-modules, the Bernstein-Sato polynomial is related with the monodromy of Milnor fibers. Let f : (Cn, 0) → (C, 0) be the germ of an analytic function such that f(0) = 0. Let Mt := f −1(t) ∩ Bǫ, where Bǫ is a ball of radius ǫ around the origin. A classical

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS 7

theorem of Milnor in the isolated singularity case, and Hamm-Le in general, state that the diffeomorphism class of Mt does not change for small enough values of ǫ and even smaller values of |t|. Fix a point t0 in C close enough to the origin. Definition 1.18. Let f : (Cn, 0) → (C, 0) be the germ of an analytic function such that f(0) = 0. The Milnor fiber of f at 0 is Ff,0 := Mt0. The cohomology vector spaces Hi(Ff,0, C) admit an action T called monodromy generated by going once around a loop starting at t0 around 0. Another classical theorem due to Landmann, Grothendieck, etc., states that the eigenvalues of the monodromy action T are roots of unity. Example 1.19. [Milnor] When f has an isolated singularity, dimC Hj(Ff,0, C) =      for j = 0, n − 1, 1 for j = 0, dimC C[[x1, . . . , xn]]/

• ∂f

∂x1, . . . , ∂f ∂xn

• for j = n − 1.

A more flexible definition of cohomology in algebraic topology is through con- structible sheaves. Recall that a constructible sheaf on a complex analytic variety X is a sheaf of finite-dimensional C-vector spaces such that there exists a stratification in the analytic topology of X with the property that the restriction of the sheaf to any element of the stratification is a locally constant sheaf (i.e. local system). Con- structible sheaves form an abelian category, that is, roughly speaking, kernels and cokernels are again constructible sheaves. Thus one can form the bounded derived category of constructible sheaves Db

c(X) consisting of complexes of sheaves F q with

constructible cohomology sheaves Hi(F q) which vanish for |i| ≫ 0, and inverting quasi-isomorphisms. The natural functors on constructible sheaves extend to de- rived functors on the derived category. For example, for a morphism p : X → Y , the direct image functor p∗ on constructible sheaves, extends to the derived direct image functor Rp∗ such that, given a short exact sequence of complexes 0 → F q

1 → F q 2 → F q 3 → 0,

• ne has a long exact sequence

. . . → Hi(Rp∗F q

1 ) → Hi(Rp∗F q 2 ) → Hi(Rp∗F q 3 ) → Hi+1(Rp∗F q 1 ) → . . .

If X is a complex analytic manifold and f : X → C is a holomorphic function. Consider the diagram f −1(0)

i

X

f

• X ×C ˜

C∗

p

• C

C∗

• ˜

C∗

q

• where ˜

C∗ is the universal cover of C∗, and p : X ×C ˜ C∗ → X is the natural

• projection. One has Deligne’s nearby cycles functor:

ψf := i∗Rp∗p∗ : Db

c(X) → Db c(f −1(0)).

8 NERO BUDUR

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 1.20. [Deligne] Let X be a complex analytic manifold and f : X → C a holomorphic function. Then Hi(Ff,x, C) = Hi(i∗

xψfCX)

and the action induced from the deck transformation of the covering ˜ C∗ → C∗ recovers the monodromy. We now explain how the geometric picture of Milnor fibers can be understood via D-modules. Let X be nonsingular complex variety of dimension n. The sheaf

• f algebraic differential operators DX is locally given in affine coordinates by the

Weyl algebra An(C). 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 regular holonomic cohomology. The topological package, consisting of the bounded derived category of con- structible sheaves Db

c(X) and the natural functors attached to it, has an algebraic

• counterpart. There is a well-defined functor

DRX : Db

rh(DX) → Db c(X),

DRXM =(Ω q

X ⊗OX M)[n], if M = DX − module,

which is an equivalence of categories commuting with the usual functors, and which probably will be covered by other lectures at this school. This is the famous Riemann-Hilbert correspondence of Malgrange, Kashiwara and Mebkhout. Thus, there is a D-module theoretic counterpart of the nearby cycles functor ψf, hence of the Milnor monodromy of a regular function f : X → C. In concrete terms, this is achieved by the V -filtration which 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 decomposition of the semisimple part of the Milnor monodromy. Let if : X → X × C be the graph embedding of f, that is, x → (x, f(x)). Theorem 1.21. [Malgrange, Kashiwara] Let X be nonsingular complex variety

• f dimension n and f : X → C a regular function. For α ∈ (0, 1],

ψf,λCX[n − 1] = DRX(Grα

V (if)+OX),

where λ = exp(−2πiα) and (if)+ is the D-module direct image. We will also explain the relation between the V -filtration and Bernstein-Sato

• polynomials. This will give the following very useful corollary for computing al-

gebraically the topological invariants attached the singularities of f in terms of Bernstein-Sato polynomials: Corollary 1.22. [Malgrange, Kashiwara] Let X be nonsingular complex variety and f : X → C a regular function.

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS 9

(a) The set V (bf) of roots of the Bernstein-Sato polynomial of f consists of negative rational numbers. (b) The set Exp (V (bf)) = {exp(2πiα) | bf(α) = 0} is equal to the set

• x∈f−1(0)
• i

{eigenvalues of the monodromy on Hi(Ff,x, C)}. The interpretation of monodromy eigenvalues and eigenspaces in terms of D- module theoretic terms allows implementable algorithms for computing these geo- metric invariants.

• 2. V -filtration

The V -filtration of Malgrange and Kashiwara is one of the cornerstones of D- modules theory because it provides a link with geometric invariants. 2.1. 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], with x = x1, . . . , xn, and t = t1, . . . , tr. So the ideal I ⊂ OY of the smooth closed subvariety X × 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 2.2. The filtration V along X × 0 on DY is V jDY := { P ∈ DY | PIi ⊂ Ii+j for all i ∈ Z }, with j ∈ Z and Ii = OY for i ≤ 0. So V jDY is generated over DX by the monomials tβ∂γ

t with |β| − |γ| ≥ j.

Remark 2.3. 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

• perators of order ≤ j, and Ij = DY,j = OY for j ≤ 0.

Let M be a finitely generated DY -module. Definition 2.4. The filtration V along X × 0 on M is an exhaustive decreasing filtration of finitely 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 Grα

V = 0, and

these α must be rational. Here, Grα

V = V α/V >α, where V >α = ∪β>αV β.

(ii) tjV α ⊂ 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 ∂tjtj − α on Grα V 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.

10 NERO BUDUR

Theorem 2.5. [Malgrange, Kashiwara] The filtration V along X × 0 on M exists if M is regular holonomic and quasi-unipotent. We have seen already what holonomic D-modules are. It suffices for our purposes to say that 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

• f existence reduces to the case r = 1.

Lemma 2.6. The filtration V along X × 0 on M is unique if it exists.

• Proof. Say

V is another filtration on M satisfying Definition 2.4. By symmetry it will be enough to show that V α ⊂ V α for every α. Suppose that α = β and consider V α ∩ V β/(V >α ∩ V β) + (V α ∩ V >β). Since both filtrations satisfy 2.4-(iv), it follows that both (

j ∂tjtj − α) and

(

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

We show now that for every α we have (4) 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 finitely many 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 (4). Since the V -filtration is discrete, a repeated application of (4) shows that for every β ≥ α we have V α ⊂ V β + V α. We deduce from 2.4-(iii) that if we fix β ≫ 0, then (5) 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 2.4-(ii), for q with q + γ ≥ α we have Iq V γ ⊂ V α. Thus IqV β ⊂ V α. Hence by (5) we have V α ⊂ V α.

• Definition 2.7. 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

• f the action of s = −

j ∂tjtj on V 0DY · m/V 1DY · m.

Proposition 2.8. [Sabbah] 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 ∂tjtj −β is nilpotent on

V β/V >β and V is indexed discretely. Then, for a given β there is a polynomial b(s)

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS 11

depending on β, having all roots ≤ −α and rational, and such that b(−

j ∂tjtj) ·

m ∈ V β. Hence it is enough to show that there is β such that V β ∩ V 0DY m ⊂ V 1DY m. 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 GrF

k A are finitely generated F0A-modules for

all k. It follows that A is also noetherian. Now

i≥0 V iM is finitely generated over A because by axiom (iii) of 2.4, there

exists i0 such that V iM is recovered from V i0M if i ≥ i0. Denote by N the V 0DY -submodule V 0DY m, 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 Gri UN is

finitely generated over

i≥0 Gri V DY . If i is big compared with the degrees of local

generators, we see that U iN ⊂ V 1DY m, which is what we wanted to show. Conversely, fix an element m ∈ 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 = αm, (

j ∂tjtj − β) is invertible on V αm/V >αm. But

bm(−

j ∂tjtj)m ∈ V >αm. Hence we must have bm(−αm) = 0.

• One can sheafify easily what we have discussed in this subsection and obtain

the corresponding statements for the V -filtration on a coherent DY -module along a smooth subvariety of Y of codimension r. 2.9. The geometry behind the V -filtration. Let X = Cn and f = (f1, . . . , fr) with fi ∈ C[x] = OX. Consider the graph embedding of f given by if : X → X × Cr = Y, x → (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 ∂ν

t = ∂ν1 t1 . . . ∂νr tr ,

g(h ⊗ ∂ν

t ) = gh ⊗ ∂ν t ,

∂xi(h ⊗ ∂ν

t ) = ∂xih ⊗ ∂ν t −

• j

∂fj ∂xi h ⊗ ∂tj∂ν

t ,

∂tj(h ⊗ ∂ν

t ) = h ⊗ ∂tj∂ν t ,

tj(h ⊗ ∂ν

t ) = fjh ⊗ ∂ν t − νjh ⊗ (∂ν t )j ,

where (∂ν

t )j is obtained from ∂ν t 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 2.5, the V -filtration on (if)+OX along X × 0

• exists. For m = h ⊗ 1 ∈ (if)+OX with h ∈ OX, we have a polynomial bm(s) as

in Definition 2.7 and the polynomials bm(s) determine the V -filtration on (if)+OX by Proposition 2.8. Remark 2.10. When r = 1, we have thus clarified what is meant by Theorem 1.21 which says that the graded pieces with respect to the V -filtration on (if)+OX admit a geometric meaning in terms of Milnor fibers. It is moreover true that the

12 NERO BUDUR

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 Tu of 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. With the notation as in 2.1, let t = (t1, . . . tr) be the last coordinates on Y = X ×Cr. There exists a specialization of Y to the normal cone NX×0Y of X ×0 in Y . More precisely, a diagram of natural maps Y Y × C∗

q

• j
• ˜

Y

ρ

• p
• NX×0Y
• C∗

C

• where the two bottom squares are cartesian (i.e.

fiber products) and the top triangle is commuting. This diagram corresponds to the diagram of natural maps

• f algebras

C[x, t]

• C[x, t, u, u−1]
• i∈Z t−iC[x, t] ⊗ ui
• i≤0 t−i/t−i+1 ⊗ ui

C[u, u−1]

• C[u]
• C[u]/u
• 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 is commuting. The effect of this is that it reduces the setup of r regular functions f = (f1, . . . , fr) to the setup of only one regular function p : ˜ Y → C. For the smooth case complete intersection case, that is, for t = (t1, . . . , tr), one can define the r > 1 analog of the Deligne nearby cycles functor, the so-called Verdier specialization functor SpX×0|Y : Db

c(Y ) → Db c(NX×0Y ),

F q → ψp(Rj∗q∗F q). For the arbitrary case f = (f1, . . . , fr) defining a closed subscheme Z ⊂ X, one defines SpZ|X : Db

c(X) → Db c(NZX),

F q → SpX×0|Y (if)∗F q. 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,λ generalizing the one for Deligne nearby cycles functor.

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS 13

By Theorem 1.21, the D-module counterpart of SpZ|X,λCX is given by Grα

V ˜

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 q ˜ M is the V -filtration along NX×0Y in ˜ Y . This V -filtration exists because ˜ M is regular holonomic and quasi-unipotent with respect to the map p. Consider now the V -filtration on M along X×0 on Y . With a bit of computation with coordinates, one can show that it is related to the V -filtration on ˜ M: V α−r+1 ˜ M =

• i∈Z

V α−iM ⊗ ui. In fact, this defines the V -filtration on M along X × 0 in Y and reduces the proof

• f its existence to the r = 1 case. This also translates into a relation between

Bernstein-Sato polynomials of m ∈ M and ˜ m = m ⊗ 1 in ˜ M with respect to the two V -filtrations: (6) b ˜

m(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 2.11. 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−i

V

(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 V (bm) of roots of the Bernstein-Sato polynomial

• f m consists of negative rational numbers.

(c) The set Exp (V (bm)) is included in the set of eigenvalues of SpZ|XCX, with equality if m generates (if)+OX.

• 3. Bernstein-Sato polynomials of varieties

3.1. Bernstein-Sato polynomials of varieties. Let K be a field of characteristic zero and let K[x] = K[x1, . . . , xn] the polynomial ring over K. Consider a collection f = (f1, . . . , fr) of polynomials fi ∈ K[x]. Then An(K) acts naturally on K[x,

r

• i=1

f −1

i

, s1, . . . , sr]

r

• i=1

f si

i .

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

• sj + 1

if i = j sj if i = j For example, ti

• j f

sj j = fi

• j f

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

14 NERO BUDUR

Definition 3.2. The Bernstein-Sato polynomial bf(s) of a collection of polynomials f = (f1, . . . , fr), fi ∈ K[x], is defined to be the non-zero monic polynomial of the lowest degree in s = r

i=1 si satisfying the relation

(7) bf(s)

r

• i=1

f si

i = r

• j=1

Pjfj

r

• i=1

f si

i ,

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

i f si i

replaced by

i f si i h.

Note that bf,1 = bf and that this agrees with the classical Bernstein-Sato poly- nomial when r = 1, cf. Definition 1.5. Example 3.3. Let f = (x2x3, x1x3, x1x2). Then bf(s) = (s + 3/2)(s + 2)2 and the

• perators sij cannot be avoided by the operators Pj in the above definition.

Lemma 3.4. Let if : X → Y = X × Cr be the graph embedding. Let m = h ⊗ 1 ∈ (if)+OX = OX[∂t]. Then bf,h(s) = bm(s).

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

s =

j sj on

DX[sij]

• j

f

sj j h/

• k

DX[sij]fk

• j

f

sj j h,

a quotient of submodules of OX[

i f −1 i

, s1, . . . , sr]

i f si i h. We can check that this

quotient is isomorphic to V 0DY m/V 1DY m. The action of tj agrees on both sides,

• j f

sj j h corresponds to m, and sj corresponds to −∂tjtj.

• By this lemma and by (6), the existence of bf(s) is reduced to that of the classical

Bernstein-Sato polynomials, that is, to the case r = 1, which we have proved. Similarly for bf,h(s). It would be interesting to give a proof along the lines for the classical case without passing through the argument via specialization to the normal cone. Theorem 2.11 gives the geometric interpretation: the exponentials of the roots

• f bf,h(s) are among the monodromy eigenvalues of SpZ|XCX. The exponentials of

the roots of bf(s) give all the eigenvalues. Theorem-Definition 3.5. [B.-Mustat ¸˘ a-Saito] Let Z denote the closed sub- scheme of An

K defined by the ideal generated by f = (f1, . . . , fr), fi ∈ K[x1, . . . , xn].

The polynomial bZ(s) := bf(s − codimXZ) depends only on Z and not on f. We call bZ(s) the Bernstein-Sato polynomial

• f the scheme Z. For non-affine schemes, bZ(s) is defined as the lowest common

multiplier of the Bernstein-Sato polynomials of the affine pieces and it is well- defined.

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS 15

3.6. Another relation with geometry: multiplier ideals. Multiplier ideals are technical tools to deal with singularities. They are one of the most important ingredients in recent advances in higher-dimensional birational algebraic geometry. We will see now how Bernstein-Sato polynomials are related with multiplier ideals. This provides algorithms for computing multiplier ideals and related invariants. Let X be a smooth complex algebraic variety or a complex manifold of dimension

• n. Let Z be any closed subscheme or closed analytic subscheme of X. Chose a

collection of local generators f = (f1, . . . , fr) of the ideal of Z. Definition 3.7. The multiplier ideal of (X, αZ) for α ∈ R>0 is locally defined as J (X, αZ) = {g ∈ OX | |g|2/(

• i

|fi|2)α is locally integrable}. In fact, there is an equivalent geometric definition. Let µ : X′ → X be a log resolution of (X, Z), that is, X has been blown up sufficiently many times such that X′ is smooth and the ideal in OX′ generated by the image of the ideal of Z is the ideal of a hypersurface H given locally by a monomial function. Let KX′/X denote the zero locus of the determinant of the Jacobian of µ. Proposition 3.8. For α ∈ R>0, J (X, αZ) = µ∗OX′(KX′/X − ⌊αH⌋), where ⌊ q⌋ 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

• f the choice of generators for the ideal of Z, and from the definition one sees that

the geometric interpretation is independent of the choice of log resolution µ. Consider now the graph embedding if : X → Y = X × Cr. Recall that (if)+OX = OX ⊗ C[∂t], where ∂t = (∂t1, . . . , ∂tr), and this DY -module has the V filtration along X × 0. Theorem 3.9. [B.-Mustat ¸˘ a-Saito] For all α > 0 and 0 < ǫ ≪ 1, J (X, (α − ǫ)Z) = (OX ⊗ 1) ∩ V α(if)+OX. In particular, by Lemma 3.4 and Proposition 2.8, we have: Corollary 3.10. (a) For all α > 0, locally J (X, αZ) = {h ∈ OX | α < c if bf,h(−c) = 0}. (b) Let the log canonical threshold lct (X, Z) be the smallest α > 0 such that J (X, αZ) = OX. Then lct (X, Z) is the negative of the biggest root of bf(s). 3.11. Challenge: generic determinantal varieties. Let Mn = Cn2 be the set

• f all complex n×n matrices. Consider the subset Zk,n ⊂ Mn consisting of matrices

with rank < k. Then Zk,n is the subvariety of Mn with ideal fk,n generated by the minors of size k of the matrix of indeterminates (xij)1≤i,j≤n. Motivated by the Monodromy Conjecture, which we will not talk about, but which relates the roots

• f Bernstein-Sato polynomials with poles of topological zeta functions, we pose:

16 NERO BUDUR

Conjecture 3.12. bfk,n(s) =

• s + n2

k s + (n − 1)2 k − 1

• . . . (s + (n − k + 1)) .

When k = n this states that the polynomial appearing in Proposition 1.2 is Bernstein-Sato polynomial of the determinant of the square matrix of indetermi- nates, which is well-known. When k = n, this example does not seem to fit into the setup of prehomogeneous vector spaces as in Proposition 1.3. It is not clear what should be the differential operators achieving the functional relation defining bfk,n, and examples are difficult to compute. One can also ask what is the Bernstein-Sato polynomial of the ideal generated by the k-minors of a rectangular matrix of indeterminates. We do not even have a conjectured formula. For the 2-minors in a 2 × 4 matrix of indeterminates, bf(s) = (s + 3)(s + 4). Other examples?

• 4. Bernstein-Sato ideals for mappings

The geometric invariants when have been talking until now generalize even fur-

• ther. One such generalization occurs frequently in topology, where one considers

cohomology support loci of local systems. The attempt to understand cohomology loci lead to an on-going investigation about Bernstein-Sato ideals for mappings. We will review definitions, state some conjectures relating Bernstein-Sato ideals and cohomology support loci, and indicate partial results. 4.1. Bernstein-Sato ideals for mappings. Let X = Cn. Let F = (f1, . . . , fr) be a collection of polynomials fj in C[x1, . . . , xn]. Let D = ∪r

i=1f −1 i

(0). We will

• ften identify F with the associated mapping F : X → Cr, as opposed to what we

did in the previous sections when we looked at the ideal generated by the fi. Definition 4.2. The Bernstein-Sato ideal of F = (f1, . . . , fr) with fi ∈ C[x1, . . . , xn] is the ideal BF generated by polynomials b ∈ C[s1, . . . , sr] such that b(s1, . . . , sr)f s1

1 · · · f sr r = Pf s1+1 1

· · · f sr+1

r

for some algebraic differential operator P ∈ C

• x1, . . . , xn,

∂ ∂x1, . . . , ∂ ∂xn, s1, . . . , sr

• .

The existence of non-zero Bernstein-Sato ideals BF can be reduced to that of the case r = 1 in Definition 2.7 and has been proved by Sabbah. In the one-variable case r = 1, the monic generator of the ideal BF is the classical Bernstein-Sato polynomial. In general, the ideal BF is not always principal. The ideal BF is generated by polynomials with coefficients in the subfield of C generated by the coefficients of F. Example 4.3. If fj are monomials, write fj = n

i=1 x ai,j i

. Let li(s1, . . . , sr) = r

j=1 ai,jsj. Let ai = r j=1 ai,j. Then

BF =

n

• i=1

(li(s) + 1) · · · (li(s) + ai).

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS 17

The following would generalize Corollary 1.22: Conjecture 4.4. (Generalized M-K property) Let F = (f1, . . . , fr) with fi ∈ C[x1, . . . , xn]. (a) The zero locus V (BF) of the Bernstein-Sato ideal of F is the zero locus of a product of linear polynomials of the form α1s1 + . . . + αrsr + α with αj ∈ Q≥0 and α ∈ Q>0. (b) Exp (V (BF)) =

• y∈D

Supp unif

y

(ψFCX). Part (a) would refine a result of Sabbah and Gyoja which states that BF,x con- tains at least one element of this type. Part (b) needs some definitions and an ex- planation as to why it would recover the Malgrange-Kashiwara property for r = 1 case. Let x ∈ X. We can similarly define the local Bernstein-Sato ideal BF,x of the germ of F at x. It is known that BF =

• x∈D

BF,x. Thus, if we let V (I) denote the zero locus of an ideal I, V (BF) =

• x∈D

V (BF,x). Moreover, this is a finite union since there is a constructible stratification of X such that for x running over a given stratum the Bernstein-Sato ideal at x is constant. Thus we can and do make a local version of the Conjecture at the point x, from which the stated version follows Let us say what has been proven about the Conjecture. Theorem 4.5. (a) Exp (V (BF,x)) ⊃

• y∈D near x

Supp unif

y

(ψFCX). (b) Supp unif

y

(ψFCX) is a finite union of torsion translated subtori of (C∗)r. (c) The set

• y∈D near x

Supp unif

y

(ψFCX) is of pure codimension one in (C∗)r. In fact, the remaining unproved portion of the Conjecture follows from:

18 NERO BUDUR

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

r

• j=1

(sj − αj)DX[s1, . . . , sr]f s1

1 . . . f sr r ≡ DX[s1, . . . , sr]f s1 1 . . . f sr r

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

1

. . . f sr+1

r

. Example 4.7. Let F = (x, y, x + y, z, x + y + z). Then the product of all entries

• f F forms a central essential indecomposable hyperplane arrangement in C3. This

means that we can easily compute the right-hand side of Conjecture 4.4. Thus, Conjecture 4.4 predicts that, in (C∗)5, (8) Exp (V (BF)) = V ((t1t2t3 − 1)(t3t4t5 − 1)(t1 . . . t5 − 1)

5

• j=1

(tj − 1)). We will check this later, see Example 4.13. However, for now note that the Bernstein-Sato ideal 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 under- standing BF however. Let DX = An(C). 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. Definition 4.9. The Bernstein-Sato ideal associated to F and M is the ideal B M

F = Bm1,...,mp F

⊂ C[s1, . . . , sr]

• f all polynomials b(s1, . . . , sr) such that

b(s1, . . . , sr)

r

• j=1

f

sj j = p

• k=1

Pk

r

• j=1

f

sj+mkj j

for some algebraic differential operators Pk in DX[s1, . . . , sr]. Remark 4.10. (a) BF, as defined before, is B 1

F , where 1 = (1, . . . , 1).

(b) For a point x in X, the local Bernstein-Sato ideal B M

F,x is similarly defined

by replacing DX with the ring DX,x of germs of holomorphic differential operators at x. Then B M

F =

• x∈X

B M

F,x.

(c) The ideals B M

F,x are non-zero by Sabbah

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS 19

Let Y = X × Cr with affine coordinates x1, . . . , xn, t1, . . . , tr. Define for m ∈ Nr V mDY := DX ⊗C

• β,γ∈Nr

β−γ≥m

Ctβ1

1 . . . tβr r ∂γ1 t1 . . . ∂γr tr

⊂ DY . The following is the D-module theoretic interpretation of Bernstein-Sato ideals and generalizes the case r = 1 from Lemma 3.4: Proposition 4.11. Let m ∈ Nr. The Bernstein-Sato ideal Bm

F consists of the

polynomials b(s1, . . . , sr) such that b(−∂t1t1, . . . , −∂trtr) · V 0DY · (1 ⊗ 1) ⊂ V mDY · (1 ⊗ 1), where 1 ⊗ 1 ∈ (if)+OX = OX[∂t], and if is the graph embedding of (f1, . . . , fr). We will denote by ek for 1 ≤ k ≤ r the vector in Nr with coordinates ekj zero if k = j and ekk = 1. Using Proposition 4.11, it is straight-forward to show the next lemma: Lemma 4.12. Let m ∈ Nr. Then

r

• j=1

B

mjej F

⊂ Bm

F ⊂ r

• j=1

B

mjej F

. Hence V (Bm

F ) = r

• j=1

V (B

mjej F

). In particular,

r

• j=1

B

ej F ⊂ BF ⊂ r

• j=1

B

ej F

and V (BF) =

r

• j=1

V (B

ej F ).

Example 4.13. This lemma is useful in practice. Let us retake Example 4.7. Recall that BF was intractable via computers. However, one can compute with dmod.lib [LM]: Be1

F = (s1 + 1)(s1 + s2 + s3 + 2)(s1 + s2 + s3 + s4 + s5 + 3),

Be2

F = (s2 + 1)(s1 + s2 + s3 + 2)(s1 + s2 + s3 + s4 + s5 + 3),

Be3

F = (s3 + 1)(s1 + s2 + s3 + 2)(s3 + s4 + s5 + 2)(s1 + s2 + s3 + s4 + s5 + 3),

Be4

F = (s4 + 1)(s3 + s4 + s5 + 2)(s1 + s2 + s3 + s4 + s5 + 3),

Be5

F = (s5 + 1)(s3 + s4 + s5 + 2)(s1 + s2 + s3 + s4 + s5 + 3).

Then (8) follows from Lemma 4.12 which says that V (BF) = 5

j=1 V (B ej F ).

20 NERO BUDUR

4.14. Cohomology support loci of local systems. Let us now define the ob- jects Supp unif

x

(ψFCX) through which the Bernstein-Sato ideals BF achieve, con- jecturally, geometric meaning. Recall that F = (f1, . . . , fr) with fj ∈ C[x1, . . . , xr], X = Cn, D = ∪r

j=1f −1 j (0),

and x ∈ D. Firstly, ψFCX is defined exactly like Deligne nearby cycles functor except now

• ne replaces C∗ with (C∗)r.

This is now a complex of A-modules, where A = C[t±1

1 , . . . , t± r ] is the affine coordinate ring of the torus 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 cohomol-

• gy A-modules Hi(ψFCX)x of the stalk at x. If some of the fj do not vanish at x,

then the corresponding monodromy is trivial and Supp x(ψFCX) lives in the smaller subtorus of S∗ which forgets the j-th coordinate. Or, in other words, the set of equations defining Supp x(ψFCX) contains the “stupid” equation tj = 1. If we for- get all the stupid equations, the result is the uniformized support Supp unif

x

(ψFCX). It has the advantage that the natural space it lives in is S∗, independent of how many functions fj do not vanish at x. In the case r = 1, there are no “stupid” equations and so this geometric picture recovers the monodromy eigenvalues of Milnor fibers, that is, Conjecture 4.4 implies Corollary 1.22. There is another, more concrete, description of the uniform support loci. For a topological space U, the space of local systems of rank one on U is identified with L(U) = Hom(H1(U, Z), C∗). Define the cohomology support locus of U to be the subset V(U) of L(U) consisting

• f local systems with non-trivial cohomology,

V(U) := {L ∈ L(U) | Hk(U, L) = 0 for some k}. Coming back to our situation, for a point x in X, let UF,x be the complement of D in a small open ball centered at x, UF,x := Ballx − (Ballx ∩ D). There is a natural embedding of L(UF,x) into the torus (C∗)r induced by F. Theorem 4.15. If the polynomials fj with fj(x) = 0 define mutually distinct reduced and irreducible hypersurface germs at x, then Supp x(ψFCX) = V(UF,x). There is absolutely no difficulty to understand the relation between Supp x(ψFCX) and cohomology support loci if the assumptions are dropped, but it becomes more involved to state it. One can similarly define the uniform cohomology support locus Vunif(UF,x) such that it agrees with Supp unif

x

(ψFCX). References

[Bj] J.-E. Bj¨

• rk, Rings of differential operators. North-Holland Mathematical Library, 21.

North-Holland Publishing Co., Amsterdam-New York, 1979. xvii+374 pp. 4

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS 21

[B2]

• N. Budur, Singularity invariants related to Milnor fibers: survey. Zeta functions in al-

gebra and geometry, 161–187, Contemp. Math., 566, Amer. Math. Soc., Providence, RI,

• 2012. 2

[LM]

• V. Levandovskyy and J. Mart´

ın Morales, dmod.lib. A Singular 3-1-3 library for alge- braic D-modules (2011). 19 KU Leuven and University of Notre Dame Current address: KU Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leu- ven, Belgium E-mail address: Nero.Budur@wis.kuleuven.be