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

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Bernstein-Sato polynomials and generalizations Nero Budur (KU Leuven) Summer school Algebra, Algorithms, and Algebraic Analysis , Rolduc Abbey, Netherlands September 2-6, 2013 Nero Budur (KU Leuven) Bernstein-Sato polynomials and

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Bernstein-Sato polynomials and generalizations

Nero Budur (KU Leuven) Summer school Algebra, Algorithms, and Algebraic Analysis, Rolduc Abbey, Netherlands September 2-6, 2013

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

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Lecture 2

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

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Geometry behind bf (s)

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

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Geometry behind bf (s) Recall:

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

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Geometry behind bf (s) Recall: For f ∈ K[x] = K[x1, . . . , xn], char(K) = 0,

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

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Geometry behind bf (s) Recall: For f ∈ K[x] = K[x1, . . . , xn], char(K) = 0, bf (s)f s = Pf s+1 for some P ∈ K[x, ∂/∂x, s].

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

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Geometry behind bf (s) Recall: For f ∈ K[x] = K[x1, . . . , xn], char(K) = 0, bf (s)f s = Pf s+1 for some P ∈ K[x, ∂/∂x, s].

  • [Malgrange, Kashiwara]:

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

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Geometry behind bf (s) Recall: For f ∈ K[x] = K[x1, . . . , xn], char(K) = 0, bf (s)f s = Pf s+1 for some P ∈ K[x, ∂/∂x, s].

  • [Malgrange, Kashiwara]: Let X be nonsingular complex variety and

f : X → C a regular function.

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

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Geometry behind bf (s) Recall: For f ∈ K[x] = K[x1, . . . , xn], char(K) = 0, bf (s)f s = Pf s+1 for some P ∈ K[x, ∂/∂x, s].

  • [Malgrange, Kashiwara]: Let X be nonsingular complex variety and

f : X → C a regular function. (a) The set V(bf ) of roots of the Bernstein-Sato polynomial of f consists of negative rational numbers.

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

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Geometry behind bf (s) Recall: For f ∈ K[x] = K[x1, . . . , xn], char(K) = 0, bf (s)f s = Pf s+1 for some P ∈ K[x, ∂/∂x, s].

  • [Malgrange, Kashiwara]: Let X be nonsingular complex variety and

f : X → C a regular function. (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}

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

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Geometry behind bf (s) Recall: For f ∈ K[x] = K[x1, . . . , xn], char(K) = 0, bf (s)f s = Pf s+1 for some P ∈ K[x, ∂/∂x, s].

  • [Malgrange, Kashiwara]: Let X be nonsingular complex variety and

f : X → C a regular function. (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)}.

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This comes from the deeper Riemann-Hilbert correspondence.

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

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This comes from the deeper Riemann-Hilbert correspondence.

  • Let X = C-analytic variety.

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

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This comes from the deeper Riemann-Hilbert correspondence.

  • Let X = C-analytic variety. F = constructible sheaf on X if

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

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This comes from the deeper Riemann-Hilbert correspondence.

  • Let X = C-analytic variety. F = constructible sheaf on X if ∃

stratification of X

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

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This comes from the deeper Riemann-Hilbert correspondence.

  • Let X = C-analytic variety. F = constructible sheaf on X if ∃

stratification of X s.t. restrictions of F to strata is a locally constant sheaf of finite rank

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

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This comes from the deeper Riemann-Hilbert correspondence.

  • Let X = C-analytic variety. F = constructible sheaf on X if ∃

stratification of X s.t. restrictions of F to strata is a locally constant sheaf of finite rank (i.e. local system).

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

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This comes from the deeper Riemann-Hilbert correspondence.

  • Let X = C-analytic variety. F = constructible sheaf on X if ∃

stratification of X s.t. restrictions of F to strata is a locally constant sheaf of finite rank (i.e. local system).

  • Db

c(X) =

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

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This comes from the deeper Riemann-Hilbert correspondence.

  • Let X = C-analytic variety. F = constructible sheaf on X if ∃

stratification of X s.t. restrictions of F to strata is a locally constant sheaf of finite rank (i.e. local system).

  • Db

c(X) = bounded derived category of constructible sheaves

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

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This comes from the deeper Riemann-Hilbert correspondence.

  • Let X = C-analytic variety. F = constructible sheaf on X if ∃

stratification of X s.t. restrictions of F to strata is a locally constant sheaf of finite rank (i.e. local system).

  • Db

c(X) = bounded derived category of constructible sheaves =

complexes of sheaves F q with constructible cohomology sheaves Hi(F q) which vanish for |i| ≫ 0

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

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This comes from the deeper Riemann-Hilbert correspondence.

  • Let X = C-analytic variety. F = constructible sheaf on X if ∃

stratification of X s.t. restrictions of F to strata is a locally constant sheaf of finite rank (i.e. local system).

  • Db

c(X) = bounded derived category of constructible sheaves =

complexes of sheaves F q with constructible cohomology sheaves Hi(F q) which vanish for |i| ≫ 0, and inverting quasi-isomorphisms.

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Natural functors on constr. sh. extend to derived functors on Db

c(X).

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Natural functors on constr. sh. extend to derived functors on Db

c(X).

E.g. p : X → Y

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

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Natural functors on constr. sh. extend to derived functors on Db

c(X).

E.g. p : X → Y gives Rp∗

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Natural functors on constr. sh. extend to derived functors on Db

c(X).

E.g. p : X → Y gives Rp∗ s.t. exact 0 → F q

1 → F q 2 → F q 3 → 0,

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

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Natural functors on constr. sh. extend to derived functors on Db

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 .. → Hi(Rp∗F q

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

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

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Natural functors on constr. sh. extend to derived functors on Db

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 .. → Hi(Rp∗F q

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

Let f : X → C holom. fc.

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

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Natural functors on constr. sh. extend to derived functors on Db

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 .. → Hi(Rp∗F q

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

Let f : X → C holom. fc. Consider f −1(0)

i

X

f

  • X ×C ˜

C∗

p

  • C

C∗

  • ˜

C∗

  • Nero Budur (KU Leuven)

Bernstein-Sato polynomials and generalizations

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Natural functors on constr. sh. extend to derived functors on Db

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 .. → Hi(Rp∗F q

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

Let f : X → C holom. fc. Consider f −1(0)

i

X

f

  • X ×C ˜

C∗

p

  • C

C∗

  • ˜

C∗

  • Nearby cycles functor

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

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Natural functors on constr. sh. extend to derived functors on Db

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 .. → Hi(Rp∗F q

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

Let f : X → C holom. fc. Consider f −1(0)

i

X

f

  • X ×C ˜

C∗

p

  • C

C∗

  • ˜

C∗

  • Nearby cycles functor ψf := i∗Rp∗p∗ : Db

c(X) → Db c(f −1(0)).

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

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Natural functors on constr. sh. extend to derived functors on Db

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 .. → Hi(Rp∗F q

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

Let f : X → C holom. fc. Consider f −1(0)

i

X

f

  • X ×C ˜

C∗

p

  • C

C∗

  • ˜

C∗

  • Nearby cycles functor ψf := i∗Rp∗p∗ : Db

c(X) → Db c(f −1(0)).

Eigenspace decomposition: ψf = ⊕λψf,λ.

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  • [Deligne]: X = C-anal. mfd., f : X → C holomorphic fc.

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  • [Deligne]: X = C-anal. mfd., f : X → C holomorphic fc. ⇒

Hi(Ff,x, C)λ = Hi(i∗

xψf,λCX).

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  • [Deligne]: X = C-anal. mfd., f : X → C holomorphic fc. ⇒

Hi(Ff,x, C)λ = Hi(i∗

xψf,λCX).

DX = sheaf of alg. diff. operators

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  • [Deligne]: X = C-anal. mfd., f : X → C holomorphic fc. ⇒

Hi(Ff,x, C)λ = Hi(i∗

xψf,λCX).

DX = sheaf of alg. diff. operators = locally An(C).

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

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  • [Deligne]: X = C-anal. mfd., f : X → C holomorphic fc. ⇒

Hi(Ff,x, C)λ = Hi(i∗

xψf,λCX).

DX = sheaf of alg. diff. operators = locally An(C). Db

rh(DX) = bdd der cat of complexes of DX-mods with regular

holonomic cohomology.

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

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  • [Deligne]: X = C-anal. mfd., f : X → C holomorphic fc. ⇒

Hi(Ff,x, C)λ = Hi(i∗

xψf,λCX).

DX = sheaf of alg. diff. operators = locally An(C). Db

rh(DX) = bdd der cat of complexes of DX-mods with regular

holonomic cohomology.

  • [Malgrange, Kashiwara, Mebkhout]:

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

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  • [Deligne]: X = C-anal. mfd., f : X → C holomorphic fc. ⇒

Hi(Ff,x, C)λ = Hi(i∗

xψf,λCX).

DX = sheaf of alg. diff. operators = locally An(C). Db

rh(DX) = bdd der cat of complexes of DX-mods with regular

holonomic cohomology.

  • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety
  • f dim n.

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  • [Deligne]: X = C-anal. mfd., f : X → C holomorphic fc. ⇒

Hi(Ff,x, C)λ = Hi(i∗

xψf,λCX).

DX = sheaf of alg. diff. operators = locally An(C). Db

rh(DX) = bdd der cat of complexes of DX-mods with regular

holonomic cohomology.

  • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety
  • f dim n. Then ∃ DRX : Db

rh(DX) ↔ Db c(X)

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

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  • [Deligne]: X = C-anal. mfd., f : X → C holomorphic fc. ⇒

Hi(Ff,x, C)λ = Hi(i∗

xψf,λCX).

DX = sheaf of alg. diff. operators = locally An(C). Db

rh(DX) = bdd der cat of complexes of DX-mods with regular

holonomic cohomology.

  • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety
  • f dim n. Then ∃ DRX : Db

rh(DX) ↔ Db c(X) s.t. if M = DX-mod, then

DRXM = (Ω q

X ⊗OX M)[n].

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

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  • [Deligne]: X = C-anal. mfd., f : X → C holomorphic fc. ⇒

Hi(Ff,x, C)λ = Hi(i∗

xψf,λCX).

DX = sheaf of alg. diff. operators = locally An(C). Db

rh(DX) = bdd der cat of complexes of DX-mods with regular

holonomic cohomology.

  • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety
  • f dim n. Then ∃ DRX : Db

rh(DX) ↔ Db c(X) s.t. if M = DX-mod, then

DRXM = (Ω q

X ⊗OX M)[n].

So ∃ D-modules counterpart of ψf :

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

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  • [Deligne]: X = C-anal. mfd., f : X → C holomorphic fc. ⇒

Hi(Ff,x, C)λ = Hi(i∗

xψf,λCX).

DX = sheaf of alg. diff. operators = locally An(C). Db

rh(DX) = bdd der cat of complexes of DX-mods with regular

holonomic cohomology.

  • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety
  • f dim n. Then ∃ DRX : Db

rh(DX) ↔ Db c(X) s.t. if M = DX-mod, then

DRXM = (Ω q

X ⊗OX M)[n].

So ∃ D-modules counterpart of ψf : via V-filtration, to define soon.

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Let if : X → X × C be x → (x, f(x)).

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

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Let if : X → X × C be x → (x, f(x)).

  • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n

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Let if : 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.

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Let if : 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

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Let if : 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], ψf,λCX[n − 1] = DRX(Grα

V(if )+OX)

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

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Let if : 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], ψf,λCX[n − 1] = DRX(Grα

V(if )+OX)

where λ = exp(−2πiα)

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

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Let if : 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], ψf,λCX[n − 1] = DRX(Grα

V(if )+OX)

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

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

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Let if : 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], ψf,λCX[n − 1] = DRX(Grα

V(if )+OX)

where λ = exp(−2πiα) and (if )+ is the D-module direct image. Next: V-filtrations, generalized Bernstein-Sato polynomials, more geometry.

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V-filtration

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

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V-filtration X = Cn,

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

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V-filtration X = Cn, DX = An(C),

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

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V-filtration X = Cn, DX = An(C), Y = X × Cr,

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

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V-filtration X = Cn, DX = An(C), Y = X × Cr, OX = C[x],

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

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V-filtration X = Cn, DX = An(C), Y = X × Cr, OX = C[x], OY = C[x, t],

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

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V-filtration X = Cn, DX = An(C), Y = X × Cr, OX = C[x], OY = C[x, t], with x = x1, . . . , xn,

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

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V-filtration X = Cn, DX = An(C), Y = X × Cr, OX = C[x], OY = C[x, t], with x = x1, . . . , xn, t = t1, . . . , tr.

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

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V-filtration X = Cn, DX = An(C), Y = X × Cr, OX = C[x], OY = C[x, t], with x = x1, . . . , xn, t = t1, . . . , tr. So the ideal I ⊂ OY of X × 0 in Y is generated by t.

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

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V-filtration X = Cn, DX = An(C), Y = X × Cr, OX = C[x], OY = C[x, t], with x = x1, . . . , xn, t = t1, . . . , tr. So the ideal I ⊂ OY of X × 0 in Y is generated by t. So DX = OX[∂x],

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

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V-filtration X = Cn, DX = An(C), Y = X × Cr, OX = C[x], OY = C[x, t], with x = x1, . . . , xn, t = t1, . . . , tr. So the ideal I ⊂ OY of X × 0 in Y is generated by t. So DX = OX[∂x], DY = OY[∂x, ∂t].

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

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V-filtration X = Cn, DX = An(C), Y = X × Cr, OX = C[x], OY = C[x, t], with x = x1, . . . , xn, t = t1, . . . , tr. So the ideal I ⊂ OY of X × 0 in Y is generated by t. So DX = OX[∂x], DY = OY[∂x, ∂t].

  • Filtration V along X × 0 on DY:

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

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V-filtration X = Cn, DX = An(C), Y = X × Cr, OX = C[x], OY = C[x, t], with x = x1, . . . , xn, t = t1, . . . , tr. So the ideal I ⊂ OY of X × 0 in Y is generated by t. So DX = OX[∂x], DY = OY[∂x, ∂t].

  • Filtration V along X × 0 on DY:

VjDY = { P ∈ DY | PIi ⊂ Ii+j for all i ∈ Z },

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

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V-filtration X = Cn, DX = An(C), Y = X × Cr, OX = C[x], OY = C[x, t], with x = x1, . . . , xn, t = t1, . . . , tr. So the ideal I ⊂ OY of X × 0 in Y is generated by t. So DX = OX[∂x], DY = OY[∂x, ∂t].

  • Filtration V along X × 0 on DY:

VjDY = { P ∈ DY | PIi ⊂ Ii+j for all i ∈ Z }, with j ∈ Z and Ii = OY for i ≤ 0.

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

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V-filtration X = Cn, DX = An(C), Y = X × Cr, OX = C[x], OY = C[x, t], with x = x1, . . . , xn, t = t1, . . . , tr. So the ideal I ⊂ OY of X × 0 in Y is generated by t. So DX = OX[∂x], DY = OY[∂x, ∂t].

  • Filtration V along X × 0 on DY:

VjDY = { P ∈ DY | PIi ⊂ Ii+j for all i ∈ Z }, with j ∈ Z and Ii = OY for i ≤ 0. So VjDY is generated over DX by the monomials tβ∂γ

t with |β| − |γ| ≥ j.

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

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  • Let M be a fin. gen. DY-module.

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  • Let M be a fin. gen. DY-module. The filtration V along X × 0 on M

is

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

slide-69
SLIDE 69
  • Let M be a fin. gen. DY-module. The filtration V along X × 0 on M

is an exhaustive decreasing filtr. of fin. gen. V0DY-submods VαM s.t.:

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

slide-70
SLIDE 70
  • Let M be a fin. gen. DY-module. The filtration V along X × 0 on M

is an exhaustive decreasing filtr. of fin. gen. V0DY-submods VαM s.t.: (i) {VαM}α is indexed left-continuously and discretely by rational numbers,

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

slide-71
SLIDE 71
  • Let M be a fin. gen. DY-module. The filtration V along X × 0 on M

is an exhaustive decreasing filtr. of fin. gen. V0DY-submods VαM s.t.: (i) {VαM}α is indexed left-continuously and discretely by rational numbers, i.e. VαM = ∩β<αVβM;

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

slide-72
SLIDE 72
  • Let M be a fin. gen. DY-module. The filtration V along X × 0 on M

is an exhaustive decreasing filtr. of fin. gen. V0DY-submods VαM s.t.: (i) {VαM}α is indexed left-continuously and discretely by rational numbers, i.e. VαM = ∩β<αVβM; (ii) (ViDY)(VαM) ⊂ Vα+iM;

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

slide-73
SLIDE 73
  • Let M be a fin. gen. DY-module. The filtration V along X × 0 on M

is an exhaustive decreasing filtr. of fin. gen. V0DY-submods VαM s.t.: (i) {VαM}α is indexed left-continuously and discretely by rational numbers, i.e. VαM = ∩β<αVβM; (ii) (ViDY)(VαM) ⊂ Vα+iM; (iii)

j tjVαM = Vα+1M for α ≫ 0;

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

slide-74
SLIDE 74
  • Let M be a fin. gen. DY-module. The filtration V along X × 0 on M

is an exhaustive decreasing filtr. of fin. gen. V0DY-submods VαM s.t.: (i) {VαM}α is indexed left-continuously and discretely by rational numbers, i.e. VαM = ∩β<αVβM; (ii) (ViDY)(VαM) ⊂ Vα+iM; (iii)

j tjVαM = Vα+1M for α ≫ 0;

(iv) the action of

j ∂tjtj − α on Grα VM = VαM/V>αM is nilpotent.

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

slide-75
SLIDE 75
  • Let M be a fin. gen. DY-module. The filtration V along X × 0 on M

is an exhaustive decreasing filtr. of fin. gen. V0DY-submods VαM s.t.: (i) {VαM}α is indexed left-continuously and discretely by rational numbers, i.e. VαM = ∩β<αVβM; (ii) (ViDY)(VαM) ⊂ Vα+iM; (iii)

j tjVαM = Vα+1M for α ≫ 0;

(iv) the action of

j ∂tjtj − α on Grα VM = VαM/V>αM is nilpotent.

  • [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

slide-76
SLIDE 76
  • Let M be a fin. gen. DY-module. The filtration V along X × 0 on M

is an exhaustive decreasing filtr. of fin. gen. V0DY-submods VαM s.t.: (i) {VαM}α is indexed left-continuously and discretely by rational numbers, i.e. VαM = ∩β<αVβM; (ii) (ViDY)(VαM) ⊂ Vα+iM; (iii)

j tjVαM = Vα+1M for α ≫ 0;

(iv) the action of

j ∂tjtj − α on Grα VM = VαM/V>αM is nilpotent.

  • [Malgrange, Kashiwara]: The filtration V along X × 0 on M exists

if M is regular holonomic and quasi-unipotent. We see later how the proof of existence reduces to the case r 1.

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

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SLIDE 77

Lemma: V qM along X × 0 is unique if it exists.

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

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SLIDE 78

Lemma: V qM along X × 0 is unique if it exists. See proof in the notes.

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

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SLIDE 79

Lemma: V qM along X × 0 is unique if it exists. See proof in the notes.

  • For m ∈ M, the Bernstein-Sato polynomial bm(s) of m is

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

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SLIDE 80

Lemma: V qM along X × 0 is unique if it exists. See proof in the notes.

  • For m ∈ M, the Bernstein-Sato polynomial bm(s) of m is the

non-zero monic minimal polynomial of the action of s = −

j ∂tjtj

  • n V0DY · m/V1DY · m.

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

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SLIDE 81

Lemma: V qM along X × 0 is unique if it exists. See proof in the notes.

  • For m ∈ M, the Bernstein-Sato polynomial bm(s) of m is the

non-zero monic minimal polynomial of the action of s = −

j ∂tjtj

  • n V0DY · m/V1DY · m.
  • [Sabbah]: If the filtration V along X × 0 on M exists, then

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

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SLIDE 82

Lemma: V qM along X × 0 is unique if it exists. See proof in the notes.

  • For m ∈ M, the Bernstein-Sato polynomial bm(s) of m is the

non-zero monic minimal polynomial of the action of s = −

j ∂tjtj

  • n V0DY · m/V1DY · m.
  • [Sabbah]: If the filtration V along X × 0 on M exists, then bm(s)

exists for all m ∈ M, and has all roots rational.

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

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SLIDE 83

Lemma: V qM along X × 0 is unique if it exists. See proof in the notes.

  • For m ∈ M, the Bernstein-Sato polynomial bm(s) of m is the

non-zero monic minimal polynomial of the action of s = −

j ∂tjtj

  • n V0DY · m/V1DY · m.
  • [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, VαM = { m ∈ M | α ≤ c if bm(−c) = 0 }.

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

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SLIDE 84

Lemma: V qM along X × 0 is unique if it exists. See proof in the notes.

  • For m ∈ M, the Bernstein-Sato polynomial bm(s) of m is the

non-zero monic minimal polynomial of the action of s = −

j ∂tjtj

  • n V0DY · m/V1DY · m.
  • [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, VαM = { m ∈ M | α ≤ c if bm(−c) = 0 }. See proof in the notes.

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

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SLIDE 85

Next: geometry behind the V-filtration.

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

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SLIDE 86

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX,

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

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SLIDE 87

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX, if : X → X × Cr = Y given by x → (x, f(x)).

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

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SLIDE 88

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX, if : X → X × Cr = Y given by x → (x, f(x)). Let t = (t1, . . . , tr) be the coords of Cr.

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

slide-89
SLIDE 89

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX, if : X → X × Cr = Y given by x → (x, f(x)). Let t = (t1, . . . , tr) be the coords of Cr. Let (if )+OX = D-mod direct image

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

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SLIDE 90

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX, if : X → X × Cr = Y given by x → (x, f(x)). Let t = (t1, . . . , tr) be the coords of Cr. Let (if )+OX = D-mod direct image = OX ⊗C C[∂t]

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

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SLIDE 91

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX, if : X → X × Cr = Y given by x → (x, f(x)). Let t = (t1, . . . , tr) be the coords of Cr. Let (if )+OX = D-mod direct image = OX ⊗C C[∂t] s.t. for g, h ∈ OX,

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

slide-92
SLIDE 92

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX, if : X → X × Cr = Y given by x → (x, f(x)). Let t = (t1, . . . , tr) be the coords of Cr. Let (if )+OX = D-mod direct image = OX ⊗C C[∂t] s.t. for g, h ∈ OX, g(h ⊗ ∂ν

t ) = gh ⊗ ∂ν t ,

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

slide-93
SLIDE 93

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX, if : X → X × Cr = Y given by x → (x, f(x)). Let t = (t1, . . . , tr) be the coords of Cr. Let (if )+OX = D-mod direct image = OX ⊗C C[∂t] s.t. for g, h ∈ OX, g(h ⊗ ∂ν

t ) = gh ⊗ ∂ν t , ∂xi(h ⊗ ∂ν t ) = ∂xih ⊗ ∂ν t − j ∂fj ∂xi h ⊗ ∂tj∂ν t ,

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

slide-94
SLIDE 94

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX, if : X → X × Cr = Y given by x → (x, f(x)). Let t = (t1, . . . , tr) be the coords of Cr. Let (if )+OX = D-mod direct image = OX ⊗C C[∂t] s.t. for g, h ∈ OX, 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 ,

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

slide-95
SLIDE 95

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX, if : X → X × Cr = Y given by x → (x, f(x)). Let t = (t1, . . . , tr) be the coords of Cr. Let (if )+OX = D-mod direct image = OX ⊗C C[∂t] s.t. for g, h ∈ OX, 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−ej t

) ,

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

slide-96
SLIDE 96

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX, if : X → X × Cr = Y given by x → (x, f(x)). Let t = (t1, . . . , tr) be the coords of Cr. Let (if )+OX = D-mod direct image = OX ⊗C C[∂t] s.t. for g, h ∈ OX, 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−ej t

) , Facts: OX = reg. holon. DX-mod

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

slide-97
SLIDE 97

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX, if : X → X × Cr = Y given by x → (x, f(x)). Let t = (t1, . . . , tr) be the coords of Cr. Let (if )+OX = D-mod direct image = OX ⊗C C[∂t] s.t. for g, h ∈ OX, 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−ej t

) , Facts: OX = reg. holon. DX-mod ⇒ (if )+OX = reg. holon. quasi-unip. DY-mod.

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

slide-98
SLIDE 98

Next: geometry behind the V-filtration. Let X = Cn, f = (f1, . . . , fr), fi ∈ C[x] = OX, if : X → X × Cr = Y given by x → (x, f(x)). Let t = (t1, . . . , tr) be the coords of Cr. Let (if )+OX = D-mod direct image = OX ⊗C C[∂t] s.t. for g, h ∈ OX, 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−ej t

) , Facts: OX = reg. holon. DX-mod ⇒ (if )+OX = reg. holon. quasi-unip. DY-mod. So ∃ V q(if )+OX along X × 0.

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

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SLIDE 99
  • When r = 1, M-K thm says ψf,λCX[n − 1] = DRX(Grα

V(if )+OX).

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

slide-100
SLIDE 100
  • When r = 1, M-K thm says ψf,λCX[n − 1] = DRX(Grα

V(if )+OX).

Moreover, s = −∂tt on V>0(if )+OX/V1(if )+OX corresponds to logarithm of unipotent part Tu of monodromy T = TsTu.

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

slide-101
SLIDE 101
  • When r = 1, M-K thm says ψf,λCX[n − 1] = DRX(Grα

V(if )+OX).

Moreover, s = −∂tt on V>0(if )+OX/V1(if )+OX corresponds to logarithm of unipotent part Tu of monodromy T = TsTu.

  • When r > 1: reduce to r = 1 via the specialization of Y to NX×0Y:

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

slide-102
SLIDE 102
  • When r = 1, M-K thm says ψf,λCX[n − 1] = DRX(Grα

V(if )+OX).

Moreover, s = −∂tt on V>0(if )+OX/V1(if )+OX corresponds to logarithm of unipotent part Tu of monodromy T = TsTu.

  • When r > 1: reduce to r = 1 via the specialization of Y to NX×0Y:

Y

q

  • j
  • ρ
  • p
  • Nero Budur (KU Leuven)

Bernstein-Sato polynomials and generalizations

slide-103
SLIDE 103
  • When r = 1, M-K thm says ψf,λCX[n − 1] = DRX(Grα

V(if )+OX).

Moreover, s = −∂tt on V>0(if )+OX/V1(if )+OX corresponds to logarithm of unipotent part Tu of monodromy T = TsTu.

  • When r > 1: reduce to r = 1 via the specialization of Y to NX×0Y:

Y × C∗

q

  • j
  • ρ
  • p
  • C∗
  • Nero Budur (KU Leuven)

Bernstein-Sato polynomials and generalizations

slide-104
SLIDE 104
  • When r = 1, M-K thm says ψf,λCX[n − 1] = DRX(Grα

V(if )+OX).

Moreover, s = −∂tt on V>0(if )+OX/V1(if )+OX corresponds to logarithm of unipotent part Tu of monodromy T = TsTu.

  • When r > 1: reduce to r = 1 via the specialization of Y to NX×0Y:

Y Y × C∗

q

  • j
  • ˜

Y

ρ

  • p
  • C∗

C

  • Nero Budur (KU Leuven)

Bernstein-Sato polynomials and generalizations

slide-105
SLIDE 105
  • When r = 1, M-K thm says ψf,λCX[n − 1] = DRX(Grα

V(if )+OX).

Moreover, s = −∂tt on V>0(if )+OX/V1(if )+OX corresponds to logarithm of unipotent part Tu of monodromy T = TsTu.

  • When r > 1: reduce to r = 1 via the specialization of Y to NX×0Y:

Y Y × C∗

q

  • j
  • ˜

Y

ρ

  • p
  • NX×0Y
  • C∗

C

  • Nero Budur (KU Leuven)

Bernstein-Sato polynomials and generalizations

slide-106
SLIDE 106

Corresponds to the maps of algebras

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

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SLIDE 107

Corresponds to the maps of 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
  • Nero Budur (KU Leuven)

Bernstein-Sato polynomials and generalizations

slide-108
SLIDE 108

Corresponds to the maps of 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 = t1, . . . , tr−i ⊂ C[x, t] for i ≤ 0 and t−i = C[x, t] for

i ≥ 0.

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

slide-109
SLIDE 109

Let M = (if )+OX

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

slide-110
SLIDE 110

Let M = (if )+OX = C[x] ⊗ C[∂t],

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

slide-111
SLIDE 111

Let M = (if )+OX = C[x] ⊗ C[∂t], ˜ M = j+q+M

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

slide-112
SLIDE 112

Let M = (if )+OX = C[x] ⊗ C[∂t], ˜ M = j+q+M = ⊕i∈ZM ⊗ ui,

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

slide-113
SLIDE 113

Let M = (if )+OX = C[x] ⊗ C[∂t], ˜ M = j+q+M = ⊕i∈ZM ⊗ ui, V q˜ M along NX×0Y in ˜ Y,

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

slide-114
SLIDE 114

Let M = (if )+OX = C[x] ⊗ C[∂t], ˜ M = j+q+M = ⊕i∈ZM ⊗ ui, V q˜ M along NX×0Y in ˜ Y, V qM along X × 0 on Y.

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

slide-115
SLIDE 115

Let M = (if )+OX = C[x] ⊗ C[∂t], ˜ M = j+q+M = ⊕i∈ZM ⊗ ui, V q˜ M along NX×0Y in ˜ Y, V qM along X × 0 on Y. Lemma: (a) Vα−r+1 ˜ M =

i∈Z Vα−iM ⊗ ui.

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

slide-116
SLIDE 116

Let M = (if )+OX = C[x] ⊗ C[∂t], ˜ M = j+q+M = ⊕i∈ZM ⊗ ui, V q˜ M along NX×0Y in ˜ Y, V qM along X × 0 on Y. Lemma: (a) Vα−r+1 ˜ M =

i∈Z Vα−iM ⊗ ui.

(b) For m ∈ M and ˜ m = m ⊗ 1 in ˜ M,

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

slide-117
SLIDE 117

Let M = (if )+OX = C[x] ⊗ C[∂t], ˜ M = j+q+M = ⊕i∈ZM ⊗ ui, V q˜ M along NX×0Y in ˜ Y, V qM along X × 0 on Y. Lemma: (a) Vα−r+1 ˜ M =

i∈Z Vα−iM ⊗ ui.

(b) For m ∈ M and ˜ m = m ⊗ 1 in ˜ M, b˜

m(s) = bm(s + r − 1).

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

slide-118
SLIDE 118

Let M = (if )+OX = C[x] ⊗ C[∂t], ˜ M = j+q+M = ⊕i∈ZM ⊗ ui, V q˜ M along NX×0Y in ˜ Y, V qM along X × 0 on Y. Lemma: (a) Vα−r+1 ˜ M =

i∈Z Vα−iM ⊗ ui.

(b) For m ∈ M and ˜ m = m ⊗ 1 in ˜ M, b˜

m(s) = bm(s + r − 1).

Gives geometrization of V for r > 1: see Thereom 2.11 in lecture notes.

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