Bernstein-Sato polynomials and generalizations Nero Budur (KU - PowerPoint PPT Presentation

� � � � � Natural functors on constr. sh. extend to derived functors on D b c ( X ) . E.g. p : X → Y gives Rp ∗ s.t. exact 0 → F q 1 → F q 2 → F q 3 → 0 , gives exact long sequence .. → H i ( Rp ∗ F q 1 ) → H i ( Rp ∗ F q 2 ) → H i ( Rp ∗ F q 3 ) → H i + 1 ( Rp ∗ F q 1 ) → .. Let f : X → C holom. fc. Consider p f − 1 ( 0 ) � � � X X × C ˜ C ∗ i f C ∗ ˜ � � C C ∗ • Nearby cycles functor ψ f := i ∗ Rp ∗ p ∗ : D b c ( X ) → D b c ( f − 1 ( 0 )) . Eigenspace decomposition: ψ f = ⊕ λ ψ f ,λ . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. • [Malgrange, Kashiwara, Mebkhout]: Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety of dim n. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety of dim n. Then ∃ DR X : D b rh ( D X ) ↔ D b c ( X ) Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety of dim n. Then ∃ DR X : D b rh ( D X ) ↔ D b c ( X ) s.t. if M = D X -mod, then DR X M = (Ω q X ⊗ O X M )[ n ] . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety of dim n. Then ∃ DR X : D b rh ( D X ) ↔ D b c ( X ) s.t. if M = D X -mod, then DR X M = (Ω q X ⊗ O X M )[ n ] . So ∃ D -modules counterpart of ψ f : Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety of dim n. Then ∃ DR X : D b rh ( D X ) ↔ D b c ( X ) s.t. if M = D X -mod, then DR X M = (Ω q X ⊗ O X M )[ n ] . So ∃ D -modules counterpart of ψ f : via V -filtration, to define soon. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Let i f : X → X × C be x �→ ( x , f ( x )) . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n and f : X → C a regular function. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n and f : X → C a regular function. For α ∈ ( 0 , 1 ] , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n and f : X → C a regular function. For α ∈ ( 0 , 1 ] , ψ f ,λ C X [ n − 1 ] = DR X ( Gr α V ( i f ) + O X ) Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n and f : X → C a regular function. For α ∈ ( 0 , 1 ] , ψ f ,λ C X [ n − 1 ] = DR X ( Gr α V ( i f ) + O X ) where λ = exp ( − 2 π i α ) Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n and f : X → C a regular function. For α ∈ ( 0 , 1 ] , ψ f ,λ C X [ n − 1 ] = DR X ( Gr α V ( i f ) + O X ) where λ = exp ( − 2 π i α ) and ( i f ) + is the D -module direct image. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n and f : X → C a regular function. For α ∈ ( 0 , 1 ] , ψ f ,λ C X [ n − 1 ] = DR X ( Gr α V ( i f ) + O X ) where λ = exp ( − 2 π i α ) and ( i f ) + is the D -module direct image. Next: V -filtrations, generalized Bernstein-Sato polynomials, more geometry. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Y = X × C r , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . So D X = O X [ ∂ x ] , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . So D X = O X [ ∂ x ] , D Y = O Y [ ∂ x , ∂ t ] . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . So D X = O X [ ∂ x ] , D Y = O Y [ ∂ x , ∂ t ] . • Filtration V along X × 0 on D Y : Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . So D X = O X [ ∂ x ] , D Y = O Y [ ∂ x , ∂ t ] . • Filtration V along X × 0 on D Y : V j D Y = { P ∈ D Y | PI i ⊂ I i + j for all i ∈ Z } , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . So D X = O X [ ∂ x ] , D Y = O Y [ ∂ x , ∂ t ] . • Filtration V along X × 0 on D Y : V j D Y = { P ∈ D Y | PI i ⊂ I i + j for all i ∈ Z } , with j ∈ Z and I i = O Y for i ≤ 0. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . So D X = O X [ ∂ x ] , D Y = O Y [ ∂ x , ∂ t ] . • Filtration V along X × 0 on D Y : V j D Y = { P ∈ D Y | PI 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 . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• Let M be a fin. gen. D Y -module. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, i.e. V α M = ∩ β<α V β M ; Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, i.e. V α M = ∩ β<α V β M ; (ii) ( V i D Y )( V α M ) ⊂ V α + i M ; Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, i.e. V α M = ∩ β<α V β M ; (ii) ( V i D Y )( V α M ) ⊂ V α + i M ; j t j V α M = V α + 1 M for α ≫ 0; (iii) � Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, i.e. V α M = ∩ β<α V β M ; (ii) ( V i D Y )( V α M ) ⊂ V α + i M ; j t j V α M = V α + 1 M for α ≫ 0; (iii) � j ∂ t j t j − α on Gr α V M = V α M / V >α M is nilpotent. (iv) the action of � Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, i.e. V α M = ∩ β<α V β M ; (ii) ( V i D Y )( V α M ) ⊂ V α + i M ; j t j V α M = V α + 1 M for α ≫ 0; (iii) � j ∂ t j t j − α on Gr α V M = V α M / V >α M is nilpotent. (iv) the action of � • [Malgrange, Kashiwara]: The filtration V along X × 0 on M exists if M is regular holonomic and quasi-unipotent. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, i.e. V α M = ∩ β<α V β M ; (ii) ( V i D Y )( V α M ) ⊂ V α + i M ; j t j V α M = V α + 1 M for α ≫ 0; (iii) � j ∂ t j t j − α on Gr α V M = V α M / V >α M is nilpotent. (iv) the action of � • [Malgrange, Kashiwara]: The filtration V along X × 0 on M exists if M is regular holonomic and quasi-unipotent. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations We see later how the proof of existence reduces to the case r 1.

Lemma: V q M along X × 0 is unique if it exists. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. • For m ∈ M , the Bernstein-Sato polynomial b m ( s ) of m is Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. • 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 . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. • 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 . • [Sabbah]: If the filtration V along X × 0 on M exists, then Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. • 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 . • [Sabbah]: If the filtration V along X × 0 on M exists, then b m ( s ) exists for all m ∈ M, and has all roots rational. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. • 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 . • [Sabbah]: 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 } . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. • 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 . • [Sabbah]: 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 } . See proof in the notes. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , ∂ f j g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , ∂ x i ( h ⊗ ∂ ν t ) = ∂ x i h ⊗ ∂ ν ∂ x i h ⊗ ∂ t j ∂ ν t − � t , j Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , ∂ f j g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , ∂ x i ( h ⊗ ∂ ν t ) = ∂ x i h ⊗ ∂ ν ∂ x i h ⊗ ∂ t j ∂ ν t − � t , j ∂ t j ( h ⊗ ∂ ν t ) = h ⊗ ∂ t j ∂ ν t , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , ∂ f j g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , ∂ x i ( h ⊗ ∂ ν t ) = ∂ x i h ⊗ ∂ ν ∂ x i h ⊗ ∂ t j ∂ ν t − � t , j t − ν j h ⊗ ( ∂ ν − e j ∂ t j ( h ⊗ ∂ ν t ) = h ⊗ ∂ t j ∂ ν t , t j ( h ⊗ ∂ ν t ) = f j h ⊗ ∂ ν ) , t Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , ∂ f j g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , ∂ x i ( h ⊗ ∂ ν t ) = ∂ x i h ⊗ ∂ ν ∂ x i h ⊗ ∂ t j ∂ ν t − � t , j t − ν j h ⊗ ( ∂ ν − e j ∂ t j ( h ⊗ ∂ ν t ) = h ⊗ ∂ t j ∂ ν t , t j ( h ⊗ ∂ ν t ) = f j h ⊗ ∂ ν ) , t Facts: O X = reg. holon. D X -mod Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , ∂ f j g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , ∂ x i ( h ⊗ ∂ ν t ) = ∂ x i h ⊗ ∂ ν ∂ x i h ⊗ ∂ t j ∂ ν t − � t , j t − ν j h ⊗ ( ∂ ν − e j ∂ t j ( h ⊗ ∂ ν t ) = h ⊗ ∂ t j ∂ ν t , t j ( h ⊗ ∂ ν t ) = f j h ⊗ ∂ ν ) , t Facts: O X = reg. holon. D X -mod ⇒ ( i f ) + O X = reg. holon. quasi-unip. D Y -mod. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , ∂ f j g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , ∂ x i ( h ⊗ ∂ ν t ) = ∂ x i h ⊗ ∂ ν ∂ x i h ⊗ ∂ t j ∂ ν t − � t , j t − ν j h ⊗ ( ∂ ν − e j ∂ t j ( h ⊗ ∂ ν t ) = h ⊗ ∂ t j ∂ ν t , t j ( h ⊗ ∂ ν t ) = f j h ⊗ ∂ ν ) , t Facts: O X = reg. holon. D X -mod ⇒ ( i f ) + O X = reg. holon. quasi-unip. D Y -mod. So ∃ V q ( i f ) + O X along X × 0. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• When r = 1, M-K thm says ψ f ,λ C X [ n − 1 ] = DR X ( Gr α V ( i f ) + O X ) . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

• When r = 1, M-K thm says ψ f ,λ C X [ n − 1 ] = DR X ( Gr α V ( i f ) + O X ) . Moreover, s = − ∂ t t on V > 0 ( i f ) + O X / V 1 ( i f ) + O X corresponds to logarithm of unipotent part T u of monodromy T = T s T u . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations