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( x ij ) be the determinant of a n × n matrix of indeterminates. Then ( s + 1)( s + 2) . . . ( s + n ) f s = det( ∂/∂x ij ) 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 �→ t ρ ( 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 ∗ ( ∂/∂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 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 [ x 1 , . . . , x n ] be a polynomial with coefficients in a field K of characteristic zero. The Bernstein-Sato polynomial of f is the non-zero monic polynomial b f ( 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 of Sato’s proof for semi-invariants of prehomogeneous vector spaces. Secondly, by a result of B. Malgrange and M. Kashiwara, the roots of b f ( 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 [ x 1 , . . . , x n ] and s ∈ C . For Re ( s ) > 0 define a locally integrable function on R n � f ( x ) s if f ( x ) > 0 , f s + ( 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 b f ( s ) . Proof. As a distribution, f s + is defined by its value on smooth compactly supported functions φ , � � f s R n φ ( x ) f s + , φ � = + dx, which converges and defines a holomorphic distribution for Re ( s ) > 0 . Now, for Re ( s ) > 0, � � R n φ ( x ) f s φ ( x ) b ( s ) f s dx b ( s ) + dx = f> 0 � R n 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 , define β a β ( x, s ) ∂x the adjoint operator � ∂ � β � P ( s ) ∗ = ( − 1) | β | a β ( x, s ) . ∂x β Integrating by parts we obtain � � R n φ ( x )( P ( s ) f s +1 ) + dx = R n P ( s ) ∗ ( φ ( x )) f s +1 dx. + So, for Re ( s ) > 0, 1 � f s b ( s ) � f s +1 , P ( s ) ∗ ( φ ) � . + , 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 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 function in the variable s . Let M be the left A n ( 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 A n ( K ( s )) action on M is defined by � ∂ j g + sg∂ j ( f ) f − 1 � ∂ j ( gf s ) = 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 (2). To see this, consider the decreasing filtration 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 (2). 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 F p A n ( K ) = Span K { x α ∂ β | | α | + | β | ≤ p } . The associated graded vector space 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 2 n variables over K . 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. The following is not too difficult to show: Lemma 1.8. M is a finitely generated left A n ( K ) -module iff M admits a good filtration.
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