BERNSTEIN-SATO POLYNOMIALS (LECTURE NOTES, UPC BARCELONA, 2015)

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1 BERNSTEIN-SATO POLYNOMIALS (LECTURE NOTES, UPC BARCELONA, 2015) NERO BUDUR Abstract. These are lecture notes based on a series lectures at the Summer school Multiplier ideals, test ideals, and Bernstein-Sato polynomials, UPC Barcelona, September 7-10, The main scope of these notes is to give some answers to the questions: What is the geometry behind the Bernstein-Sato polynomials? What can you do with them? Contents 2 1. Classical Bernstein-Sato polynomials Origins Proof of existence Example: hyperplane arrangements The geometry behind classical Bernstein-Sato polynomials Bernstein-Sato polynomials of ideals and varieties Bernstein-Sato polynomials for ideals Bernstein-Sato polynomials of schemes Example: generic determinantal varieties Relation with multiplier ideals V -filtration Riemann-Hilbert correspondence V -filtrations on D-modules Bernstein-Sato polynomials of sections of D-modules The geometry behind the V -filtration Bernstein-Sato ideals Bernstein-Sato ideals for many functions Ideals of Bernstein-Sato type Relation with local systems Proof of Part (a) of Theorem References 25 Date: September 6, This work was partially supported by an FWO grant, a KU Leuven OT grant, and a Flemish Methusalem grant. 1

2 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, They correct and go further than an earlier version used for a series of lecture at the Summer school Algebra, Algorithms, and Algebraic Analysis, Rolduc Abbey, Netherlands, September 2-6, The main scope of these notes is to give some answers to the questions: What is the geometry 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. For a 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(x ij ) be the determinant of a n n matrix (x ij ) 1 i,j n of indeterminates. Then (1.2.1) (s + 1)(s + 2)... (s + n)f s = det( / x ij )f s+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 fundamental 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 complement 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 t ρ(g) 1 makes (G, V, ρ ) into a prehomogeneous vector space as well. One can show that for a semiinvariant f of (G, V, ρ) associated to the character χ, f (y) = f(ȳ) 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 prehomogeneous 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 ( / x 1,..., / x n )f(x) s+1. Example 1.4. Let G = GL(n, C) act on the space V = M n (C) of complex n n matrices via the usual multiplication ρ(g) : x 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), and Proposition 1.3 generalizes Proposition 1.2. Definition 1.5. Let f K[x 1,..., x n ] be a polynomial with coefficients in a field K of characteristic zero. The Bernstein-Sato polynomial of f, also called the b-function, is the 2

3 non-zero monic polynomial b f (s) of minimal degree among those b K[s] such that (1.5.1) for some operator P K[x, / x, s]. b(s)f s = P f s+1 Remark 1.6. T. Oaku [Oa-97] has used the theory of Gröbner bases to develop and implement the first algorithm for computing Bernstein-Sato polynomials. Nowadays, the computer programs Macaulay2, Risa/Asir, Singular, have packages and commands allowing the computation of Bernstein-Sato polynomials. Remark 1.7. One can define Bernstein-Sato polynomials of a (germ of an) analytic function f : C n C by replacing the ring of polynomials with the ring of (germs of) analytic functions, and then allowing the operator P to have analytic coefficients instead of polynomial coefficients in x. Remark 1.8. It is non-trivial that non-zero Bernstein-Sato polynomials exist. The existence was proved by I.N. Bernstein, independently of Sato s proof for semi-invariants of prehomogeneous vector spaces. 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[x 1,..., x n ] and s C. For Re(s) > 0 define a locally integrable function on R n { f(x) f+(x) s 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, 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 1.9. As a distribution, f s + admits a meromorphic continuation with poles in the set A N, where A is the set of roots of b f (s). Proof. As a distribution, f+ s is defined by its value on smooth compactly supported functions φ, f+, s φ = φ(x)f+dx, s R n which converges and defines a holomorphic distribution for Re(s) > 0. Now, for Re(s) > 0, b(s) φ(x)f+dx s = φ(x)b(s)f s dx R n f>0 = φ(x)(p (s)f s+1 ) + dx, R n where b(s)f s = P (s)f s+1 as in Definition 1.5. If P (s) = β a β(x, s) ( β, x) define the adjoint operator P (s) = ( ) β ( 1) β a β (x, s). x β 3

4 Integrating by parts we obtain So, for Re(s) > 0, φ(x)(p (s)f s+1 ) + dx = R n R n P (s) (φ(x))f s+1 + dx. f+, s φ = 1 s+1 f+, P (s) (φ). b(s) 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 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-79]. The proof of the existence of a non-zero Bernstein-Sato polynomial of a (germ of an) analytic function will not be covered here. For a field of characteristic zero K, let A n (K) = K[x, ] be the Weyl algebra, that is, the non-commutative ring of algebraic differential operators with x = x 1,..., x n, i = / x i, = 1,..., n, and the usual relations i x j x j i = δ ij. Let f K[x] be a non-constant polynomial. Let s be a dummy variable and K(s) the field 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 a natural left A n (K(s))-module, where the left A n (K(s)) action is defined (as expected) by j (gf s ) = ( j g + sg j (f)f 1) f s, x j (gf s ) = x j gf s, for g K(s)[x, f 1 ]. If we can show that M has finite length as a left A n (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 the decreasing filtration of M by A n (K(s))-submodules A n (K(s)) f v f s, for v = 1, 2,... By the finite length assumption, there is w Z >0 such that R(s)f w+1 f s = f w f s for some R(s) A n (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 x α 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 A n (K(s))-module is a consequence of M being a holonomic A n (K(s))-module. We keep the notation simple and work from now with a left A n (K)-module M. To explain what holonomicity is, we first explain why M being a finitely generated A n (K)-module is equivalent to M admitting a special kind of filtration. On A n (K) there is the increasing Bernstein filtration F of K-vector spaces defined by The associated graded vector space F p A n (K) = Span K {x α β α + β p}. Gr F A n (K) = p F p /F p 1 is a graded commutative ring due to the fact that F p F q F p+q. In fact, Gr F A n (K) is isomorphic with the polynomial ring in 2n variables over K. 4

5 A filtration F on M is a filtration of K-vector spaces such that p F p M = M and F p A n (K) F q M F p+q M. In this case, one has an associated graded Gr F A n (K)-module Gr F M, and we say the F is a good filtration if Gr F (M) is a finitely generated Gr F A n (K)-module. The following is not too difficult to show: Lemma M is a finitely generated left A n (K)-module iff M admits a good filtration. Now we explain what it means for a finitely generated left A n (K)-module M to be of finite length. Since dim K F p M = i p dim K 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, for some a j Q and p 0. We let dim K F p M = a d p d a 0 d(m) = d, e(m) = d!a d. 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 Let F and F be two filtrations on a left A n (K)-module M. Assume F is good. Then there exists q such that F p F p+q for all p. Proof. Since F is good, there exists an integer p 0 such that Gr F M is generated over Gr F A n (K) by elements of degree p 0. Let q be such that F p0 M F qm. One can show then that F p M F p+qm for all p. A fundamental result is Bernstein s inequality: Theorem (Bernstein) If M is a non-zero finitely generated left A n (K)-module, then d(m) n. Definition A non-zero finitely generated left A n (K)-module M is holonomic if d(m) = n. Proposition Every strictly increasing sequence of A n (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 increasing. This follows from the property that under short exact sequences of A n (K)-modules (1.15.1) 0 M 1 M 2 M 3 0, d(m 2 ) = max{d(m 1 ), d(m 3 )} and e(m 2 ) = e(m 1 ) + e(m 3 ). bounded by e(m). However, the multiplicity is There is a useful numerical criterion to guarantee that a module is holonomic: Proposition Let M be a left A n (K)-module. If F is a filtration on M and if there exist positive integers c 1 and c 2 such that dim K F p M c 1 p n + c 2 (p + 1) n 1 for all p, then M is holonomic. Proof. Let M 0 be a finitely generated submodule of M. Let F be a good filtration on M 0. M 0 has also the induced filtration F from M. By Lemma 1.12, we can find q such that F pm 0 F p+q M 0 for all p. Then dim K F pm 0 dim K F p+q M c 1 (p+q) n +c 2 (p+q +1) n 1 c 1 p n + c 3 (p + 1) n 1, for a new constant c 3. This implies that d(m 0 ) n, hence d(m 0 ) = n 5

6 and e(m 0 ) n!c 1. So, every finitely generated A n (K)-submodule of M is holonomic and has length n!c 1 over A n (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 Let f K[x] = K[x 1,..., x n ]. Let M be the left A n (K(s))-module K(s)[x, f 1 ]f s. Then M is holonomic over A n (K(s)). Proof. Let F p M = Span K(s) {gf p f s g K(s)[x], deg x g (deg f + 1)p}. Then dim K(s) F p M (deg f + 1) n p n /n! + c 2 (p + 1) n 1 for some constant c 2. One also checks that F forms a filtration. Then, by Proposition 1.16, M is holonomic over A n (K(s)) 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[x 1,..., x n ] defines a hyperplane arrangement in C n 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 of 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. Theorem (M. Saito) The roots of the Bernstein-Sato polynomial b f (s) of a central essential hyperplane arrangement f with d = deg f lie in the interval ( 2 + 1/d, 0), and the multiplicity of the root s = 1 is n. Theorem (U. Walther) With same assumptions as above, if the hyperplanes forming f are in general position and d = deg f > n, then 2d 2 ( b f (s) = (s + 1) n 1 s + k ). d Conjecture (Budur-Mustaţă-Teitler [BMZ-11]) Let f be an indecomposable essential central hyperplane arrangement in C n of degree d. Then b f ( n/d) = 0. The importance of this conjecture lies in the fact that if true, it would prove the Strong Monodromy 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 Conjecture 1.21 holds for: (a) (Budur-Saito-Yuzvinsky [BSY-11]) Reduced f with n 3, and for some other special cases; (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 (U. Walther [Wa-15]) The Bernstein-Sato polynomial of a hyperplane arrangement is not a combinatorial invariant. 6 k=n

7 An invariant is said to be combinatorial if it only depends on the lattice of intersection of the hyperplanes together with the codimensions of these intersections, and it does not depend on the position of the hyperplanes. Here is U. Walther s example, worked out completely by M. Saito too: Example ([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-Sato polynomials differ by one root: 4 16 b f (s) = (s + 1) (s + i/3) (s + i/9), b g (s) = (s + 1) i=2 4 (s + i/3) i=2 i=3 15 i=3 (s + i/9) The geometry behind classical Bernstein-Sato polynomials. Definition Let f : (C n, 0) (C, 0) be the germ of an analytic function such that f(0) = 0. The Milnor fiber of f at 0 is F f,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 F f,0 is well-defined as a diffeomorphism class. The cohomology vector spaces H i (F f,0, C) admit an action called the monodromy generated by going once around a loop around 0 C. Another classical theorem due to Landmann, Grothendieck, etc., is: Theorem (Monodromy Theorem) Let f : (C n, 0) (C, 0) be the germ of an analytic function such that f(0) = 0. The eigenvalues of the monodromy action on H (F f,0, C) are roots of unity. Example (Milnor) When f has an isolated singularity, 0 for j 0, n 1, dim C H j (F f,0, C) = 1 ( ) for j = 0, f dim C C[[x 1,..., x n ]]/ x 1,..., f x n for j = n 1. The last dimension is the Milnor number of f and can be computed by the computer programs mentioned earlier. The following classical result of Malgrange and Kashiwara shows the geometric content behind the Bernstein-Sato polynomials. Theorem (Malgrange, Kashiwara) Let f : (C n, 0) (C, 0) be the germ of an analytic function. Then: 7

8 (a) The set Zero(b f ) of roots of the Bernstein-Sato polynomial of f consists of negative rational numbers. (b) The set Exp (Zero(b f )) = {exp(2πiα) b f (α) = 0} is equal to the set {eigenvalues of the monodromy on H i (F f,x, C)}. x f 1 (0) i Z x close to 0 Remark Note that this result also implies the Monodromy Theorem. In particular, the Bernstein-Sato polynomials provide a way to use computers to calculate the set of eigenvalues 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 and let K[x] = K[x 1,..., x n ] the polynomial ring over K. Consider an ideal I K[x] together with a set of generators I = f 1,..., f r K[x] with f i K[x]. Then A n (K) acts naturally on K[x, r i=1 f 1 i, s 1,..., s r ] Define an A n (K)-linear action t i for i = 1,..., r by { sj + 1 if i = j t i (s j ) = if i j This is applied to s j appearing as powers f s j j Let s ij = s i t 1 i t j for i, j {1,..., r}. s j r i=1 f s i i. as well. For example, t i j f s j j = f i j f s j j. Definition 2.2. (Budur-Mustaţă-Saito [BMS-06a]) The Bernstein-Sato polynomial b I (s) of the ideal I = f 1,..., f r K[x] is defined to be the non-zero monic polynomial b I (s) K[s] of the lowest degree in s = r i=1 s i among those b(s) K[s] satisfying the relation r r r (2.2.1) b(s) f s i i = P j f j f s i i, for some P j A n (K)[s ij ] 1 i,j, r. For h K[x], we define similarly with i f s i i replaced by i f s i i h. i=1 j=1 b I,h (s) K[s] Remark 2.3. One can prove that the definition of b I,h is independent of the choice of generators for the ideal I. Note that b I,1 (s) = b I (s). Also, b f,1 (s) = b f (s) is the classical Bernstein-Sato polynomial defined when r = 1, cf. Definition 1.5. Remark 2.4. We will see later that the existence of generalized Bernstein-Sato polynomials can be reduced to the hypersurface case. In particular, the roots of b I (s) are all negative rational numbers as well. 8 i=1

9 Remark 2.5. The first algorithm implemented for calculation of Bernstein-Sato polyomials b I,h was due to T. Shibuta [Sh-11], building on the more precise description of b I,h from [BMS-06a] and on T. Oaku s algorithm [Oa-97]. Example 2.6. Let I = x 2 x 3, x 1 x 3, x 1 x 2. Then b I (s) = (s + 3/2)(s + 2) 2 and the operators s ij with i j cannot be avoided by the operators P j in the above definition. Remark 2.7. More generally, the roots of the Bernstein-Sato polynomials b I (s) for monomial ideals I have been determined combinatorially, in two ways, in [BMS-06b, BMS-06c]. It is interesting to remark that [BMS-06c] uses positive characteristic techniques. However, the multiplicities of these roots were not addressed so far, and this remains an open question Bernstein-Sato polynomials of schemes. Let I = f 1,..., f r K[x] be an ideal. Let Z denote the closed subscheme of A n K defined by the ideal I. Let c be the codimension of Z in A n K. Theorem 2.9. ([BMS-06a]) The polynomial depends only on Z and not on I. b Z (s) := b I (s c) Definition We call b Z (s) the Bernstein-Sato polynomial of the scheme Z. For nonaffine schemes Z, b Z (s) is defined as the lowest common multiplier of the Bernstein-Sato polynomials of the affine pieces; one can show this is well-defined Example: generic determinantal varieties. This is actually a challange, rather than an example. The idea is to generalize the formula from Cayley s example for the Bernstein-Sato polynomial of the determinant of a generic square matrix. Let M = M m,n = C mn be the set of all complex m n matrices. Consider the subset Z k = Z k,m,n M consisting of matrices with rank < k. Then Z k is the subvariety of M with ideal I k generated by the minors of size k of the matrix of indeterminates (x ij ) 1 i m;1 j n. The basic question is to find formula for the Bernstein-Sato polynomial of Z k, or equivalently, of I k. Define k 1 c k (s) = i=0 ( s + ) (m i)(n i). k i The roots of the polynomial c k (s) are the poles of the Igusa-Denef-Loeser zeta function of I k, by work of Roi Docampo [Do-13]. The proof in [Do-13] is for the case m = n, but he has informed us that the case m n follows by a similar proof. Therefore the Strong Monodromy Conjecture implies that c k (s) divides the Bernstein-Sato polynomial b Ik (s). Around 2009, we stated a conjecture: if m = n, then c k (s) = b Ik (s). When k = m = n this states that the polynomial appearing in Proposition 1.2 is Bernstein-Sato polynomial of the determinant of the square matrix of indeterminates, which is well-known. The case k n = m does not seem to fit into the setup of prehomogeneous vector spaces as in Proposition 1.3. The conjecture was disproved around 2013 by a computer calculation: 9

10 Example (T. Oaku) Let m = n = 3, k = 2, so that I 2 is the ideal of 2-by-2 minors of the matrix x 11 x 12 x 13 x 21 x 22 x 23. x 31 x 32 x 33 T. Oaku has computed that The factor s + 5 does not appear in c 2 (s). b I2 (s) = (s + 9/2)(s + 4)(s + 5). Example Let k = 2, m = 2, n = 3, so that I 2 (s) is generated by the 2-minors in the matrix ( ) x11 x 12 x 13. x 21 x 22 x 23 In this case, b I2 (s) = (s + 2)(s + 3) = c 2 (s). Example Let k = 2, m = 2, n = 4, so that I 2 (s) is generated by the 2-minors in the matrix ( ) x11 x 12 x 13 x 14. x 21 x 22 x 23 x 24 In this case, b I2 (s) = (s + 3)(s + 4) = c 2 (s). Example (Raicu-Walther-Weyman) Let k = n m, so that I n is the ideal generated by the maximal minors of the generic m-by-n matrix of indeterminates. Then b In (s) is a divisor of the polynomial c k (s) = (s + m)... (s + m n + 1). Example (T. Shibuta) In the example above, the equality b In (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 I k that computers can handle Relation with multiplier ideals. The standard reference for multiplier ideal is the textbook [La-04]. 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 local generators I Z = f 1,..., f r of the ideal of Z. Definition The multiplier ideal of (X, αz) for α R >0 is locally defined as { } g 2 J (X, αz) = g O X ( i f is locally integrable. i 2 ) α 10

11 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 O X generated by the image of the ideal of Z is the ideal of a hypersurface H given locally by a monomial function. Let K X /X denote the zero locus of the determinant of the Jacobian of µ. Proposition For α R >0, J (X, αz) = µ O X (K X /X αh ), where 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 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. If J (X, αz) J (X, (α ɛ)z) for 0 < ɛ 1, we call α a jumping number of (X, Z). The smallest jumping number is called the log canonical threshold and it is denoted lct (X, Z). Hence lct (X, Z) is the smallest α > 0 such that J (X, αz) O X. Theorem ([BMS-06a]) (a) For all α > 0, locally J (X, αz) = {h O X α < c if b I,h ( c) = 0}. (b) lct (X, Z) is the biggest root of b I ( s). (c) Any jumping number of (X, Z) in the interval [lct (X, Z), lct (X, Z) + 1) is a root of b I ( s). Remark 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 1.29 and 2.20, nor we have said anything about why non-zero Bernstein-Sato polynomials of type b I,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 the geometry behind the Bernstein-Sato polynomials of ideals. 3. V -filtration The underlying concept used in linking Bernstein-Sato polynomials with geometry is the concept of V -filtrations. We introduce this concept in this section indirectly: via the Riemann-Hilbert correspondence. The reason is that this correspondence provides the geometry behind the whole theory of D-modules. The V -filtration can be seen as a particular case of explicit Riemann-Hilbert correspondence. In this section we will also sketch the proofs of Theorems 1.29, 2.20, along with the existence of b I,h (s), we will give the geometry behind them, and we will finish with the mentioning that the multiplier-ideal filtrations are restrictions of V -filtrations. 11

12 3.1. Riemann-Hilbert correspondence. A more flexible definition of cohomology in algebraic topology is through constructible 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). Constructible 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 Dc(X) b consisting of complexes of sheaves F with constructible cohomology sheaves H i (F ) which vanish for i 0, and inverting quasi-isomorphisms. The natural functors on constructible sheaves extend to derived 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 1 F 2 F 3 0, one has a long exact sequence... H i (Rp F 1 ) H i (Rp F 2 ) H i (Rp F 3 ) H i+1 (Rp F 1 )... If X is a complex analytic manifold and f : X C is a holomorphic function. Consider the diagram f 1 (0) p X X C C i f C 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: C q C ψ f := i Rp p : D b c(x) D b c(f 1 (0)). Let C X be the constant sheaf on X. If x is a point such that f(x) = 0, i x : {x} f 1 (0) is the natural inclusion, and F f,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 holomorphic function. Then H i (F f,x, C) = H i (i xψ f C X ) and the action induced from the deck transformation of the covering C C recovers the monodromy 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 algebraic linear differential operators D X is locally given in affine coordinates by the Weyl algebra A n (C). If X is analytic, one takes D X to be the sheaf of analytic linear differential operators. An important class of (left) D X -modules consists of those regular and holonomic. We have already introduced holonomicity, but we will not say more about the regularity. Let Drh b (D X) be the bounded derived category of complexes of D X -modules with regular holonomic cohomology. 12

13 The topological package, consisting of the bounded derived category of constructible sheaves D b c(x) and the natural functors attached to it, has a D-theoretic counterpart. There is a well-defined functor, called the de Rham functor, DR X : D b rh(d X ) D b c(x), that is an equivalence of categories commuting with the usual functors, and such that it restrict to an equivalence between the abelian category of regular holonomic D-modules and that of perverse sheaves on X. This is the famous Riemann-Hilbert correspondence of Kashiwara and Mebkhout. Thus, there is a D-module counterpart of the nearby cycles functor ψ f, hence of the Milnor monodromy for a 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 ψ f = λ ψ f,λ, with λ roots of unity, be the functor decomposition corresponding to the eigenspace decomposition of the semisimple part of the Milnor monodromy. Let i f : X X C be the graph embedding of f sending x to (x, f(x)). Theorem 3.3. (Malgrange, Kashiwara) Let X be complex manifold of dimension n and f : X C a holomorphic function. For α (0, 1], ψ f,λ C X [n 1] = DR X (Gr α V (i f ) + O X ), where λ = exp( 2πiα) and (i f ) + is the D-module direct image. In the case of a regular function on a nonsingular algebraic variety, one should view this as relating on one hand classical singularity theory, represented on the left-hand side by the Milnor fiber monodromy via the theorem above due to Deligne, with, on the other hand, 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 some explicit D-module. We give more details about the V -filtration next V -filtrations on D-modules. Let X = C n be the complex affine n-space. Let D X be the Weyl algebra A n (C). Let Y = X C r. Denote by O X = C[x], O Y = C[x, t], with x = x 1,..., x n, and t = t 1,..., t r. So the ideal I O Y of the smooth closed subvariety X 0 of Y is generated by t. With this notation, D X = O X [ x ], D Y = O Y [ x, t ], with x = x1,... xn, xi = / x i, and similarly for t. We will consider only left D-modules. Definition 3.5. The filtration V along X 0 on D Y V j D Y := { P D Y P I i I i+j for all i Z }, with j Z and I i = O Y for i 0. So V j D Y is generated over D X by the monomials t β γ t with β γ j. Remark 3.6. By computation with local coordinates, one can show: (i) V j 1 D Y V j 2 D Y V j 1+j 2 D Y, with equality if j 1, j 2 0; (ii) V j D Y = I j V 0 D Y D Y, j = D Y, j V 0 D Y I j, where D Y,j D Y order j, and I j = D Y,j = O Y for j is are the operators of

14 Let M be a finitely generated D Y -module. Definition 3.7. The filtration V along X 0 on M is an exhaustive decreasing filtration of finitely generated V 0 D Y -submodules V α := V α M, such that: (i) {V α } α is indexed left-continuously and discretely by rational numbers, i.e. V α = β<α V β, every interval contains only finitely many α with Gr α V 0, and these α must be rational. Here, Gr α V = V α /V >α, where V >α = β>α V β. (ii) t j V α V α+1, and tj V α V α 1 for all α Q, i.e. (V i D Y )(V α M) V α+i M; (iii) j t jv α = V α+1 for α 0; (iv) the action of j t j t j α on Gr α V is nilpotent. All conditions depend only on the variety Y = X C r 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 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 of existence reduces to the case r = 1. Lemma 3.9. The filtration V along X 0 on M is unique if it exists. Proof. Say Ṽ is another filtration on M satisfying Definition 3.7. By symmetry it will be enough to show that V α Ṽ α for every α. Suppose that α β and consider V α Ṽ β /(V >α Ṽ β ) + (V α Ṽ >β ). Since both filtrations satisfy 3.7-(iv), it follows that both ( j t j t j α) and ( j t j t j 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 + Ṽ α. Fix m V α. By exhaustion, there is β 0 (in particular β < α) such that m Ṽ β. By what we have already proved, we may write m = m 1 + m 2, with m 1 V >α and m 2 V α Ṽ >β. Say m 2 Ṽ β 1 with β 1 > β. If we replace m by m 2 and β by β 1, then the class in V α /V >α remains unchanged, and repeat the process. Since the filtration Ṽ 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 Ṽ 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 + Ṽ α. We deduce from 3.7-(iii) that if we fix β 0, then (3.9.2) V α I q V β + Ṽ α for q N. By coherence, V β = V 0 D Y m i for finitely many m i. By exhaustion, there exists some γ Z such that Ṽ γ contains the m i, hence also V β. By 3.7-(ii), for q with q + γ α we have I q Ṽ γ Ṽ α. Thus I q V β Ṽ α. Hence by (3.9.2) we have V α Ṽ α. 14

15 3.10. Bernstein-Sato polynomials of sections of D-modules. We keep the notation as above. Definition Let M be a finitely generated D Y -module. For m M, the Bernstein- Sato polynomial b m (s) of m is the non-zero monic minimal polynomial of the action of s = j t j t j on V 0 D Y m/v 1 D Y m. Proposition (Sabbah [Sa-84]) If the filtration V along X 0 on M exists, then b m (s) exists for all m M, and has all roots rational. Moreover, V α M = { m M α c if b m ( c) = 0 }. Proof. Suppose that the filtration V exists on M. Let m M. Then m V α M for some α, since V is an exhaustive filtration. Recall that j t j t j 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, having all roots α and rational, and such that b( j t j t j ) m V β. Hence it is enough to show that there is β such that V β V 0 D Y m V 1 D Y m. Let A = i 0 V i D Y τ i and define F k A = i 0 (V i D Y D Y,k )τ i. Then F 0 A and Gr F A are noetherian rings and Gr F k A are finitely generated F 0 A-modules for all k. It follows that A is also noetherian. Now i 0 V i M is finitely generated over A because by axiom (iii) of 3.7, there exists i 0 such that V i M is recovered from V i 0 M if i i 0. Denote by N the V 0 D Y -submodule V 0 D Y m, and let U i = V i N for i 0. Then i 0 U i N 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 D Y. If i is big compared with the degrees of local generators, we see that U i N V 1 D Y m, which is what we wanted to show. Conversely, fix an element m M and suppose that α c whenever b m ( c) = 0. Let α m = max{β m V β }. We need to show that α α m. It is enough to show that b m ( α m ) = 0. For β α m, ( j t j t j β) is invertible on V αm /V >αm. But b m ( j t j t j )m V >αm. Hence we must have b m ( α m ) = 0. One can sheafify easily what we have discussed in this subsection and obtain the corresponding statements for the V -filtration on a coherent D Y -module along a smooth subvariety of Y of codimension r The geometry behind the V -filtration. Let X = C n and f = (f 1,..., f r ) with f i C[x] = O X. Consider the graph embedding of f given by i f : X X C r = Y, x (x, f 1 (x),..., f r (x)). Let t = (t 1,..., t r ) be the coordinates of C r. Let (i f ) + O X be the D-module direct image. By definition, this means that (i f ) + O X = O X C C[ t1,..., tr ] with the left D Y -action given as follows: for g, h O X, and ν t = ν 1 t 1... νr t r, g(h ν t ) = gh ν t, xi (h ν t ) = xi h ν t j f j x i h tj ν t, tj (h ν t ) = h tj ν t, t j (h ν t ) = f j h ν t ν j h ( ν t ) j, 15

16 where ( ν t ) j is obtained from ν t by replacing ν j with ν j 1. Two facts need to be mentioned: O X is a regular holonomic D X -module, and the direct image (i f ) + of such a D X -module is a regular holonomic quasi-unipotent D Y -module. Thus, by Theorem 3.8, the V -filtration on (i f ) + O X along X 0 exists. For m = h 1 (i f ) + O X with h O X, we have a polynomial b m (s) as in Definition 3.11 and the polynomials b m (s) determine the V -filtration on (i f ) + O X by Proposition Remark When r = 1, we have thus clarified what is meant by the right-hand side the equality in Theorem 3.3 which says that the graded pieces with respect to the V -filtration on (i f ) + O X admit a geometric meaning in terms of Milnor fibers. It is moreover true that the action of monodromy is reflected in the following fashion in D-module theoretic terms: s = t t on V >0 (i f ) + O X /V 1 (i f ) + O X corresponds to the logarithm of the unipotent part T u of the monodromy T on ψ f C X under the Jordan decomposition T = T s T u. Let us say now what geometric meaning is behind the V -filtration in the case r > 1. With the notation as in 3.4, let t = (t 1,... t r ) be the last coordinates on Y = X C r. There exists a specialization of Y to the normal cone N X 0 Y of X 0 in Y. More precisely, a diagram of natural maps Y q Y C j p Ỹ ρ N X 0 Y C C 0 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 of algebras C[x, t] C[x, t, u, u 1 ] i Z t i C[x, t] u i i 0 t i /t i+1 u i C[u, u 1 ] C[u] C[u]/u where t i is the ideal t 1,..., t r i in O Y = 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 = (f 1,..., f r ) to the setup of only one regular function p : Ỹ C. For the smooth complete intersection case, that is, for t = (t 1,..., t r ), one can define the r > 1 analog of the Deligne nearby cycles functor, the so-called Verdier specialization functor Sp X 0 Y : D b c(y ) D b c(n X 0 Y ), F ψ p (Rj q F ). For the arbitrary case f = (f 1,..., f r ) defining a closed subscheme Z X, one defines Sp Z X : D b c(x) D b c(n Z X), F Sp X 0 Y (i f ) F. 16

17 In fact, for r = 1 this recovers the Deligne nearby cycles functor. Taking into account the monodromy from the map p : Ỹ C, there is a monodromy action on Sp Z X with eigenvalues roots of unity, and a decomposition Sp Z X = λ Sp Z X,λ generalizing the one for Deligne nearby cycles functor. By Theorem 3.3, the D-module counterpart of Sp Z X,λ C X is given by Gr α M V where α (0, 1], λ = exp( 2πiα), M = (i f ) + O X = C[x] C[ t ] with the natural action of D Y, M = j + q + M = i Z M u i with the natural action of D Y C[u, u ], and V M is the V -filtration along N X 0 Y in Ỹ. 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 = V α i M u i. i Z In fact, this defines the V -filtration on M along X 0 in Y and reduces the proof of 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: (3.14.1) b m (s) = b m (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 ([BMS-06a]) Let X be a nonsingular complex variety. (a) Let f = (f 1,..., f r ) define a closed subscheme Z in X with f i functions. For α (0, 1], ( ) Sp Z X,λ C X = DR NX Y Gr α i V (i f ) + O X u i i Z : X C regular up to a shift, where λ = exp( 2πiα) and V is the filtration along X 0 on (i f ) + O X. (b) Let m (i f ) + O X. The set Zero(b m ) of roots of the Bernstein-Sato polynomial of m consists of negative rational numbers. (c) The set Exp (Zero(b m )) is included in the set of eigenvalues of Sp Z X C X, with equality if m generates (i f ) + O X. This implies and generalizes Theorem This theorem provides the existence, rationality, and the geometry behind the more general Bernstein-Sato polynomials b I,h (s) for ideals via the following lemma. Let I = f 1,..., f r. Lemma Let m = h 1 (i f ) + O X = O X [ t ]. Then b I,h (s) = b m (s). Proof. It is enough to show that b m (s) is the minimal polynomial of the action of s = j s j on D X [s ij ] f s j j h/ D X [s ij ]f k f s j j h, j k j 17

18 a quotient of submodules of O X [ i f 1 i, s 1,..., s r ] i f s i i h. We can check that this quotient is isomorphic to V 0 D Y m/v 1 D Y m. The action of t j agrees on both sides, j f s j j h corresponds to m, and s j corresponds to tj t j. Remark Indeed, by this lemma and by (3.14.1), the existence of b I (s) is reduced to that of the classical Bernstein-Sato polynomials, that is, to the case r = 1, which we have proved. Similarly for b I,h (s). It would be interesting to give a proof of the existence of b I,h (s) along the lines for the classical case without passing through the argument via specialization to the normal cone. Theorem 3.15 gives the geometric interpretation of b I,h (s), generalizing Theorem 1.29: Corollary ([BMS-06a]) The exponentials of the roots of b I,h (s) are among the monodromy eigenvalues of Sp Z X C X. The exponentials of the roots of b I (s) give all the eigenvalues. Theorem 3.15 also implies, via the previous lemma, Theorem 2.20 since we have the following D-module theoretic interpretation of multiplier ideals: Theorem ([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 < ɛ 1, J (X, (α ɛ)z) = (O X 1) V α (i f ) + O X. Remark One should interpret this result, in case Z is a hypersurface given by a function f, as relating the multiplier ideals with classical singularity theory, represented on 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 formalism of 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] Bernstein-Sato ideals for many functions. Let X = C n. Let F = (f 1,..., f r ) be a collection of polynomials f j in C[x 1,..., x n ]. Let D = r i=1f 1 i (0). We will often identify F with the associated mapping F : X C r, as opposed to what we did in the previous sections when we looked at the ideal generated by the f i. Let D X = A n (C). Definition 4.2. The Bernstein-Sato ideal of F = (f 1,..., f r ) with f i C[x 1,..., x n ] is the ideal B F generated by polynomials b C[s 1,..., s r ] such that b(s 1,..., s r )f s 1 1 fr sr = P f s fr sr+1 for some algebraic differential operator P D X [s 1,..., s r ]. The existence of non-zero Bernstein-Sato ideals B F is similar to that of the case r = 1 in Definition 3.11 as shown by Lichtin [Li-88]. One can similarly define the local Bernstein-Sato ideal B F,x of a finite collection of germs of holomorphic functions at a point in x C n by using analytic differential operators. The proof of the existence in this case is due to Sabbah [Sa-87], see also [Ba-05]. 18

19 In the one-variable case r = 1, the monic generator of the ideal B F is the classical Bernstein-Sato polynomial. In general, the ideal B F is not always principal. The ideal B F is generated by polynomials with coefficients in the subfield of C generated by the coefficients of F. Example 4.3. If f j are monomials, write f j = n i=1 xa i,j i. Let l i (s 1,..., s r ) = r j=1 a i,js j. Let a i = r j=1 a i,j. Then n B F = (l i (s) + 1) (l i (s) + a i ). i=1 The following would generalize Theorem 1.29: Conjecture 4.4. ([Bu-15]) Let F = (f 1,..., f r ) with f i C[x 1,..., x n ]. (a) The Bernstein-Sato ideal B F is generated by products of linear polynomials of the form α 1 s α r s r + α with α j Q 0 and α Q >0. (b) Exp (Zero(B F )) = y D Supp y (ψ F C X ). Here, we let Exp : (C ) r C r be the map α exp(2πα), and we let Zero(I) denote the zero locus of an ideal I. The complex ψ F C X in Dc(X), b which we call the Sabbah specialization complex, is defined exactly like Deligne nearby cycles complex except now one 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 t j denote monodromies with respect to the f j. Now, Supp x (ψ F C X ) denotes the union of all the supports in S of the cohomology A-modules H i (ψ F C X ) x of the stalk at x. In the case r = 1, this geometric picture recovers the monodromy eigenvalues of Milnor fibers, that is, Conjecture 4.4 implies Corollary Part (a) would refine a result of Sabbah [Sa-90] and Gyoja [Gy-93] which states that B F,x contains at least one element of this type. Part (b) needs some definitions and an explanation as to why it would recover the Malgrange-Kashiwara property for r = 1 case. Let x X. It is known that B F = B F,x. x D Thus, Zero(B F ) = x D Zero(B F,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. With notation as above, we have: 19

20 (a) ([Bu-15]) Exp (Zero(B F,x )) y D near x Supp y (ψ F C X ), and equality is conjectured. 1 (b) (Budur-Wang [BW-15b]) Supp y (ψ F C X ) 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 portion of the Conjecture would follow from: Conjecture 4.6. Let x X. Assume that F = (f 1,..., f r ) is such that the f j with f j (x) = 0 define mutually distinct reduced and irreducible hypersurface germs at x. Then, locally at x, for all α Zero(B F,x ), r (s j α j )D X [s 1,..., s r ]f s fr sr D X [s 1,..., s r ]f s fr sr j=1 modulo D X [s 1,..., s r ]f s 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 forms a central essential indecomposable hyperplane arrangement in C 3, the cone over these lines: This means that we can easily compute the right-hand side of Conjecture 4.4, which then predicts that, in (C ) 5, 5 (4.7.1) Exp (Zero(B F )) = Zero( (t 1 t 2 t 3 1)(t 3 t 4 t 5 1)(t 1... t 5 1) (t j 1) ). We will check this later, see Example However, for now note that the Bernstein-Sato ideal B F of F is currently intractable via computer 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 B F however. Let F = (f 1,..., f r ) with f j C[x 1,..., x n ]. Let M = {m k N r k = 1,..., p} be a collection of vectors, which we also view as an p r matrix M = (m kj ) with m kj = (m k ) j. 1 As pointed out by Liu-Maxim [LM-14], in all the statements in [Bu-15] where the uniform support Supp unif ψ F C X of the Sabbah specialization complex appears, the unif should be dropped to conform to what is proven in [Bu-15]. 20 j=1

21 Definition 4.9. The Bernstein-Sato ideal associated to F and M is the ideal B M F of all polynomials b(s 1,..., s r ) such that r b(s 1,..., s r ) f s j j = = B m 1,...,m p F C[s 1,..., s r ] for some algebraic differential operators P k in D X [s 1,..., s r ]. j=1 p k=1 P k r j=1 f s j+m kj j Remark (a) B F, as defined before, is BF 1, where 1 = (1,..., 1). (b) For a point x in X, the local Bernstein-Sato ideal BF,x M is similarly defined by replacing D X with the ring D X,x of germs of holomorphic differential operators at x. Then B M F = x X B M F,x. (c) The ideals BF,x M are non-zero by Sabbah. Theorem ([Bu-15]) Let m N r. For j = 1,..., r, let t j be the ring isomorphism of C[s 1,..., s r ] defined by t j (s i ) = s i + δ ij. Then there are inclusions of ideals in C[s 1,..., s r ] 1 j r m j >0 m j 1 k=0 t m t m j 1 j 1 tk j B e j F Bm F 1 j r m j >0 m j 1 k=0 t m t m j 1 j 1 tk j B e j F. Here δ ij = 0 if i j, and δ ii = 1. Also, we denote by e j the r-tuple with the k-th entry δ jk. By convention, t 0 j is the identity map, the product map t a t ar r means the obvious composition of maps, and t a t ar r I is the image of the ideal I under this product map. Example This result is useful in practice. Let us retake Example 4.7. Recall that B F was intractable via computers. However, one can compute with dmod.lib [LM-11]: B e 1 F = (s 1 + 1)(s 1 + s 2 + s 3 + 2)(s 1 + s 2 + s 3 + s 4 + s 5 + 3), B e 2 F = (s 2 + 1)(s 1 + s 2 + s 3 + 2)(s 1 + s 2 + s 3 + s 4 + s 5 + 3), B e 3 F = (s 3 + 1)(s 1 + s 2 + s 3 + 2)(s 3 + s 4 + s 5 + 2)(s 1 + s 2 + s 3 + s 4 + s 5 + 3), B e 4 F = (s 4 + 1)(s 3 + s 4 + s 5 + 2)(s 1 + s 2 + s 3 + s 4 + s 5 + 3), B e 5 F = (s 5 + 1)(s 3 + s 4 + s 5 + 2)(s 1 + s 2 + s 3 + s 4 + s 5 + 3). Then (4.7.1) follows from Theorem 4.11 which implies in particular that 5 Exp (V (B F )) = Exp (V (B e j F )). j=1 Theorem 4.11 is a consequence of the following result, in which we will use the notation t m = r j=1 tm j j. Lemma Let m, n N r. Then there are inclusions of ideals B m F (t m B n F ) B m+n F B m F (t m B n F ). 21

22 Proof. We will use the notation f s = r j=1 f s j j. Let b 1 B m F and b 2 B n F. Write b 1f s = P 1 f s+m and b 2 f s = P 2 f s+n for some P 1 and P 2 in D X [s 1,..., s r ]. Apply t m to both sides of b 2 f s = P 2 f s+n. We obtain then that (t m b 2 )f s+m = (t m P 2 )f s+m+n. Applying P 1 on the left on both sides of the equality, we have P 1 (t m P 2 )f s+m+n = P 1 (t m b 2 )f s+m = (t m b 2 )P 1 f s+m = (t m b 2 )b 1 f s. Thus b 1 (t m b 2 ) is in B m+n F, which implies the first inclusion. Take now b B m+n F. Write bf s = P f s+m+n for some P in D X [s 1,..., s r ]. Then bf s = P f n f s+m, so b BF m. Now, multiply by f m on the left on both sides of the last equality. We obtain that bf s+m = f m P f s+m+n. This shows that b t m BF n. Hence b Bm F (tm BF n), which proves the second inclusion Relation with local systems. There is another, more concrete, description of the support loci Supp y (ψ F C X ) of the Sabbah specialization complex in terms of rank one local systems. For a topological space U, the space M B (U) of local systems of rank one on U is a group under the tensor product. It is identified by mondromies around loops with M B (U) = Hom(H 1 (U, Z), C ) = (C ) r (finite abelian group). Define the cohomology jump loci of U and the cohomology support loci to be Σ i k(u) := {L M B (U) dim H i (U, L) k}, Σ(U) := {L M B (U) H (U, L) 0} = Σ i 1(U). i Coming back to our situation, for a point x in X, let U F,x be the complement of D in a small open ball centered at x, There is a natural pullback map U F,x := Ball x (Ball x D). F : M B ((C ) r ) = (C ) r M B (U F,x ) induced by F. This is an isomorphism if the polynomials f j define the mutually distinct reduced and irreducible analytic branches of f = j f j at x. Theorem Supp x (ψ F C X ) = (F ) 1 (Σ(U F,x )). The Part (b) of Theorem 4.5 is a particular case of the recent more difficult result: Theorem (Budur-Wang [BW-15b]) Σ i k (U F,x) is finite union of torsion translated subtori in M B (U F,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 Is there a refined Bernstein-Sato type ideal B i k C[s 1,..., s r ], depending on F and x, such that Exp (Zero(B i k)) = Σ i k(u F,x )? 22