MotivationThe sheaf model

Blow up techniqueThe joint kernel

Blow up technique for a class of Hilbert modules

Shibananda Biswas , BGU

(joint work with Gadadhar Misra)

Tel Aviv UniversityTrilateral Operator Algebras/Operator Theory Seminar

December 15, 2010

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

The Cowen - Douglas class

A Hilbert module over the polynomial ring C[z] := C[z1, . . . , zm] is aHilbert space H which is a C[z] -module with the assumption

‖p · f‖ ≤ Cp‖f‖, f ∈ H, p ∈ C[z],

for some Cp > 0.

The multiplication Mj by the coordinate functions zj ,Mjf := zj · f, 1 ≤ j ≤ m, then defines a commutative tupleM = (M1, ...,Mm) of linear bounded operators acting on H andvice-versa. Let Ω be a bounded domain in Cm .

A Hilbert module H over the polynomial ring C[z] is said to be in theCowen-Douglas class Bn(Ω) , n ∈ N , if

dim H/mwH = n <∞ for all w ∈ Ω

∩w∈ΩmwH = 0, where mw denotes the maximal ideal in C[z] at w.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

The Cowen - Douglas class

A Hilbert module over the polynomial ring C[z] := C[z1, . . . , zm] is aHilbert space H which is a C[z] -module with the assumption

‖p · f‖ ≤ Cp‖f‖, f ∈ H, p ∈ C[z],

for some Cp > 0.

The multiplication Mj by the coordinate functions zj ,Mjf := zj · f, 1 ≤ j ≤ m, then defines a commutative tupleM = (M1, ...,Mm) of linear bounded operators acting on H andvice-versa. Let Ω be a bounded domain in Cm .

A Hilbert module H over the polynomial ring C[z] is said to be in theCowen-Douglas class Bn(Ω) , n ∈ N , if

dim H/mwH = n <∞ for all w ∈ Ω

∩w∈ΩmwH = 0, where mw denotes the maximal ideal in C[z] at w.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

The Cowen - Douglas class

A Hilbert module over the polynomial ring C[z] := C[z1, . . . , zm] is aHilbert space H which is a C[z] -module with the assumption

‖p · f‖ ≤ Cp‖f‖, f ∈ H, p ∈ C[z],

for some Cp > 0.

The multiplication Mj by the coordinate functions zj ,Mjf := zj · f, 1 ≤ j ≤ m, then defines a commutative tupleM = (M1, ...,Mm) of linear bounded operators acting on H andvice-versa. Let Ω be a bounded domain in Cm .

A Hilbert module H over the polynomial ring C[z] is said to be in theCowen-Douglas class Bn(Ω) , n ∈ N , if

dim H/mwH = n <∞ for all w ∈ Ω

∩w∈ΩmwH = 0, where mw denotes the maximal ideal in C[z] at w.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

The Cowen - Douglas class

A Hilbert module over the polynomial ring C[z] := C[z1, . . . , zm] is aHilbert space H which is a C[z] -module with the assumption

‖p · f‖ ≤ Cp‖f‖, f ∈ H, p ∈ C[z],

for some Cp > 0.

The multiplication Mj by the coordinate functions zj ,Mjf := zj · f, 1 ≤ j ≤ m, then defines a commutative tupleM = (M1, ...,Mm) of linear bounded operators acting on H andvice-versa. Let Ω be a bounded domain in Cm .

A Hilbert module H over the polynomial ring C[z] is said to be in theCowen-Douglas class Bn(Ω) , n ∈ N , if

dim H/mwH = n <∞ for all w ∈ Ω

∩w∈ΩmwH = 0, where mw denotes the maximal ideal in C[z] at w.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

The Cowen - Douglas class

A Hilbert module over the polynomial ring C[z] := C[z1, . . . , zm] is aHilbert space H which is a C[z] -module with the assumption

‖p · f‖ ≤ Cp‖f‖, f ∈ H, p ∈ C[z],

for some Cp > 0.

The multiplication Mj by the coordinate functions zj ,Mjf := zj · f, 1 ≤ j ≤ m, then defines a commutative tupleM = (M1, ...,Mm) of linear bounded operators acting on H andvice-versa. Let Ω be a bounded domain in Cm .

A Hilbert module H over the polynomial ring C[z] is said to be in theCowen-Douglas class Bn(Ω) , n ∈ N , if

dim H/mwH = n <∞ for all w ∈ Ω

∩w∈ΩmwH = 0, where mw denotes the maximal ideal in C[z] at w.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

The Cowen - Douglas class

A Hilbert module over the polynomial ring C[z] := C[z1, . . . , zm] is aHilbert space H which is a C[z] -module with the assumption

‖p · f‖ ≤ Cp‖f‖, f ∈ H, p ∈ C[z],

for some Cp > 0.

The multiplication Mj by the coordinate functions zj ,Mjf := zj · f, 1 ≤ j ≤ m, then defines a commutative tupleM = (M1, ...,Mm) of linear bounded operators acting on H andvice-versa. Let Ω be a bounded domain in Cm .

A Hilbert module H over the polynomial ring C[z] is said to be in theCowen-Douglas class Bn(Ω) , n ∈ N , if

dim H/mwH = n <∞ for all w ∈ Ω

∩w∈ΩmwH = 0, where mw denotes the maximal ideal in C[z] at w.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

The Cowen - Douglas class (continued)

Let T = (T1, . . . , Tm) be an m -tuple of bounded linear operators onthe Hilbert space K .

Let DT : K→ K⊗ Cm be the operator defined byDTf = (T1f, . . . , Tmf), f ∈ K .

The operator tuple T is said to be in the Cowen-Douglas classBn(Ω) , n ∈ N , if

ranDT−w is closed for all w ∈ Ω

spankerDT−w : w ∈ Ω is dense in K

dim kerDT−w = n for all w ∈ Ω

A Hilbert module H over the polynomial ring C[z] is in Bn(Ω) if andonly if the adjoint of the multiplication operator tupleM∗ = (M∗1 , ...,M

∗m) is in Bn(Ω∗) , where Ω∗ := z : z ∈ Ω .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

The Cowen - Douglas class (continued)

Let T = (T1, . . . , Tm) be an m -tuple of bounded linear operators onthe Hilbert space K .

Let DT : K→ K⊗ Cm be the operator defined byDTf = (T1f, . . . , Tmf), f ∈ K .

The operator tuple T is said to be in the Cowen-Douglas classBn(Ω) , n ∈ N , if

ranDT−w is closed for all w ∈ Ω

spankerDT−w : w ∈ Ω is dense in K

dim kerDT−w = n for all w ∈ Ω

A Hilbert module H over the polynomial ring C[z] is in Bn(Ω) if andonly if the adjoint of the multiplication operator tupleM∗ = (M∗1 , ...,M

∗m) is in Bn(Ω∗) , where Ω∗ := z : z ∈ Ω .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

The Cowen - Douglas class (continued)

Let T = (T1, . . . , Tm) be an m -tuple of bounded linear operators onthe Hilbert space K .

Let DT : K→ K⊗ Cm be the operator defined byDTf = (T1f, . . . , Tmf), f ∈ K .

The operator tuple T is said to be in the Cowen-Douglas classBn(Ω) , n ∈ N , if

ranDT−w is closed for all w ∈ Ω

spankerDT−w : w ∈ Ω is dense in K

dim kerDT−w = n for all w ∈ Ω

A Hilbert module H over the polynomial ring C[z] is in Bn(Ω) if andonly if the adjoint of the multiplication operator tupleM∗ = (M∗1 , ...,M

∗m) is in Bn(Ω∗) , where Ω∗ := z : z ∈ Ω .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

Examples

A Hilbert module H in Bn(Ω) determines a holomorphic Hermitianvector bundle on EH on Ω∗ defined by

EH := (w, f) ∈ Ω∗ ×H : f ∈ kerD(M−w)∗ and π(w, f) = w.

Cowen and Douglas prove that isomorphic Hilbert modulescorrespond to equivalent vector bundles and vice-versa.

Also, they provide a model for the Hilbert modules in Bn(Ω). Cowenand Douglas (Curto and Salinas, in general) show that these modulescan be realized as a Hilbert space consisting of holomorphic functionson Ω possessing a reproducing kernel. The module action is thensimply the pointwise multiplication.

Examples are Hardy and the Bergman modules over the ball and thepoly-disc in Cm.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

Examples

A Hilbert module H in Bn(Ω) determines a holomorphic Hermitianvector bundle on EH on Ω∗ defined by

EH := (w, f) ∈ Ω∗ ×H : f ∈ kerD(M−w)∗ and π(w, f) = w.

Cowen and Douglas prove that isomorphic Hilbert modulescorrespond to equivalent vector bundles and vice-versa.

Also, they provide a model for the Hilbert modules in Bn(Ω). Cowenand Douglas (Curto and Salinas, in general) show that these modulescan be realized as a Hilbert space consisting of holomorphic functionson Ω possessing a reproducing kernel. The module action is thensimply the pointwise multiplication.

Examples are Hardy and the Bergman modules over the ball and thepoly-disc in Cm.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

Examples

A Hilbert module H in Bn(Ω) determines a holomorphic Hermitianvector bundle on EH on Ω∗ defined by

EH := (w, f) ∈ Ω∗ ×H : f ∈ kerD(M−w)∗ and π(w, f) = w.

Cowen and Douglas prove that isomorphic Hilbert modulescorrespond to equivalent vector bundles and vice-versa.

Also, they provide a model for the Hilbert modules in Bn(Ω). Cowenand Douglas (Curto and Salinas, in general) show that these modulescan be realized as a Hilbert space consisting of holomorphic functionson Ω possessing a reproducing kernel. The module action is thensimply the pointwise multiplication.

Examples are Hardy and the Bergman modules over the ball and thepoly-disc in Cm.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

Examples

A Hilbert module H in Bn(Ω) determines a holomorphic Hermitianvector bundle on EH on Ω∗ defined by

EH := (w, f) ∈ Ω∗ ×H : f ∈ kerD(M−w)∗ and π(w, f) = w.

Cowen and Douglas prove that isomorphic Hilbert modulescorrespond to equivalent vector bundles and vice-versa.

Also, they provide a model for the Hilbert modules in Bn(Ω). Cowenand Douglas (Curto and Salinas, in general) show that these modulescan be realized as a Hilbert space consisting of holomorphic functionson Ω possessing a reproducing kernel. The module action is thensimply the pointwise multiplication.

Examples are Hardy and the Bergman modules over the ball and thepoly-disc in Cm.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

Not an example!

However, many natural examples of Hilbert modules fail to be in theclass Bn(Ω).

For instance, H20 (D2) := f ∈ H2(D2) : f(0) = 0 is not in Bn(D2).

The problem is that the dimension of the joint kernel

H/mwH ∼= ∩mj=0Ker(Mj − wj)∗ = kerD(M−w)∗

is no longer a constant.

Indeed, we have (an easy calculation)

dim(H/mwH

)=

1 ifw 6= (0, 0)

2 if w = (0, 0).

We outline an attempt to systematically study examples like the onegiven above using methods of complex analytic geometry.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

Not an example!

However, many natural examples of Hilbert modules fail to be in theclass Bn(Ω).

For instance, H20 (D2) := f ∈ H2(D2) : f(0) = 0 is not in Bn(D2).

The problem is that the dimension of the joint kernel

H/mwH ∼= ∩mj=0Ker(Mj − wj)∗ = kerD(M−w)∗

is no longer a constant.

Indeed, we have (an easy calculation)

dim(H/mwH

)=

1 ifw 6= (0, 0)

2 if w = (0, 0).

We outline an attempt to systematically study examples like the onegiven above using methods of complex analytic geometry.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

Not an example!

However, many natural examples of Hilbert modules fail to be in theclass Bn(Ω).

For instance, H20 (D2) := f ∈ H2(D2) : f(0) = 0 is not in Bn(D2).

The problem is that the dimension of the joint kernel

H/mwH ∼= ∩mj=0Ker(Mj − wj)∗ = kerD(M−w)∗

is no longer a constant.

Indeed, we have (an easy calculation)

dim(H/mwH

)=

1 ifw 6= (0, 0)

2 if w = (0, 0).

We outline an attempt to systematically study examples like the onegiven above using methods of complex analytic geometry.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

Not an example!

However, many natural examples of Hilbert modules fail to be in theclass Bn(Ω).

For instance, H20 (D2) := f ∈ H2(D2) : f(0) = 0 is not in Bn(D2).

The problem is that the dimension of the joint kernel

H/mwH ∼= ∩mj=0Ker(Mj − wj)∗ = kerD(M−w)∗

is no longer a constant.

Indeed, we have (an easy calculation)

dim(H/mwH

)=

1 ifw 6= (0, 0)

2 if w = (0, 0).

We outline an attempt to systematically study examples like the onegiven above using methods of complex analytic geometry.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

Not an example!

However, many natural examples of Hilbert modules fail to be in theclass Bn(Ω).

For instance, H20 (D2) := f ∈ H2(D2) : f(0) = 0 is not in Bn(D2).

The problem is that the dimension of the joint kernel

H/mwH ∼= ∩mj=0Ker(Mj − wj)∗ = kerD(M−w)∗

is no longer a constant.

Indeed, we have (an easy calculation)

dim(H/mwH

)=

1 ifw 6= (0, 0)

2 if w = (0, 0).

We outline an attempt to systematically study examples like the onegiven above using methods of complex analytic geometry.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

Definitions

A Hilbert module M ⊂ O(Ω) is said to be in the class B1(Ω) if

it possesses a reproducing kernel K ( we don’t rule out the possibility:K(w,w) = 0 for w in some closed subset X of Ω ) and

the dimension of M/mwM is finite for all w ∈ Ω (but not necessarilyconstant).

Most of the examples in B1(Ω) are obtained by taking submodules ofHilbert modules H(⊆ O(Ω)) in the Cowen-Douglas class B1(Ω).

Are there others?

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

Definitions

A Hilbert module M ⊂ O(Ω) is said to be in the class B1(Ω) if

it possesses a reproducing kernel K ( we don’t rule out the possibility:K(w,w) = 0 for w in some closed subset X of Ω ) and

the dimension of M/mwM is finite for all w ∈ Ω (but not necessarilyconstant).

Most of the examples in B1(Ω) are obtained by taking submodules ofHilbert modules H(⊆ O(Ω)) in the Cowen-Douglas class B1(Ω).

Are there others?

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The class Bn(Ω)The class B1(Ω)

Definitions

A Hilbert module M ⊂ O(Ω) is said to be in the class B1(Ω) if

it possesses a reproducing kernel K ( we don’t rule out the possibility:K(w,w) = 0 for w in some closed subset X of Ω ) and

the dimension of M/mwM is finite for all w ∈ Ω (but not necessarilyconstant).

Most of the examples in B1(Ω) are obtained by taking submodules ofHilbert modules H(⊆ O(Ω)) in the Cowen-Douglas class B1(Ω).

Are there others?

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

Construction of the sheaf model

Following the correspondence of a vector bundle with a locally freesheaf, we construct a sheaf SM(Ω) for the Hilbert module M.

The sheaf SM is the subsheaf of the sheaf of holomorphic functionsO(Ω) whose stalk SM

w at w ∈ Ω is(f1)wOw + · · ·+ (fn)wOw : f1, . . . , fn ∈M, n ∈ N

where Ow is the space of germs of holomorphic function at the pointw

For any Hilbert module M in B1(Ω) , the sheaf SM is coherent.

This is essentially Noether’s stationary lemma!

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

Construction of the sheaf model

Following the correspondence of a vector bundle with a locally freesheaf, we construct a sheaf SM(Ω) for the Hilbert module M.

The sheaf SM is the subsheaf of the sheaf of holomorphic functionsO(Ω) whose stalk SM

w at w ∈ Ω is(f1)wOw + · · ·+ (fn)wOw : f1, . . . , fn ∈M, n ∈ N

where Ow is the space of germs of holomorphic function at the pointw

For any Hilbert module M in B1(Ω) , the sheaf SM is coherent.

This is essentially Noether’s stationary lemma!

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

Construction of the sheaf model

Following the correspondence of a vector bundle with a locally freesheaf, we construct a sheaf SM(Ω) for the Hilbert module M.

The sheaf SM is the subsheaf of the sheaf of holomorphic functionsO(Ω) whose stalk SM

w at w ∈ Ω is(f1)wOw + · · ·+ (fn)wOw : f1, . . . , fn ∈M, n ∈ N

where Ow is the space of germs of holomorphic function at the pointw

For any Hilbert module M in B1(Ω) , the sheaf SM is coherent.

This is essentially Noether’s stationary lemma!

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

Construction of the sheaf model

Following the correspondence of a vector bundle with a locally freesheaf, we construct a sheaf SM(Ω) for the Hilbert module M.

The sheaf SM is the subsheaf of the sheaf of holomorphic functionsO(Ω) whose stalk SM

w at w ∈ Ω is(f1)wOw + · · ·+ (fn)wOw : f1, . . . , fn ∈M, n ∈ N

where Ow is the space of germs of holomorphic function at the pointw

For any Hilbert module M in B1(Ω) , the sheaf SM is coherent.

This is essentially Noether’s stationary lemma!

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

The decomposition theorem

Theorem. Suppose g0i , 1 ≤ i ≤ d, be a minimal set of generators for

the stalk SMw0 . Then there exists a open neighborhood Ω0 of w0 such

that

K(·, w) := Kw = g01(w)K(1)

w + · · ·+ g0d(w)K(d)

w , w ∈ Ω0

for some choice of anti-holomorphic functionsK(1), . . . ,K(d) : Ω0 →M ,

the vectors K(i)w , 1 ≤ i ≤ d , are linearly independent in M for w in Ω0

the vectors K(i)w0 | 1 ≤ i ≤ d are uniquely determined by these

generators g01 , . . . , g

0d ,

the linear span of the set of vectors K(i)w0 | 1 ≤ i ≤ d in M is

independent of the generators g01 , . . . , g

0d , and

the vectors K(i)w0 , 1 ≤ i ≤ d , are eigenvectors for the adjoint of the

action of C[z] on the Hilbert module M at w0 .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

The decomposition theorem

Theorem. Suppose g0i , 1 ≤ i ≤ d, be a minimal set of generators for

the stalk SMw0 . Then there exists a open neighborhood Ω0 of w0 such

that

K(·, w) := Kw = g01(w)K(1)

w + · · ·+ g0d(w)K(d)

w , w ∈ Ω0

for some choice of anti-holomorphic functionsK(1), . . . ,K(d) : Ω0 →M ,

the vectors K(i)w , 1 ≤ i ≤ d , are linearly independent in M for w in Ω0

the vectors K(i)w0 | 1 ≤ i ≤ d are uniquely determined by these

generators g01 , . . . , g

0d ,

the linear span of the set of vectors K(i)w0 | 1 ≤ i ≤ d in M is

independent of the generators g01 , . . . , g

0d , and

the vectors K(i)w0 , 1 ≤ i ≤ d , are eigenvectors for the adjoint of the

action of C[z] on the Hilbert module M at w0 .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

The decomposition theorem

Theorem. Suppose g0i , 1 ≤ i ≤ d, be a minimal set of generators for

the stalk SMw0 . Then there exists a open neighborhood Ω0 of w0 such

that

K(·, w) := Kw = g01(w)K(1)

w + · · ·+ g0d(w)K(d)

w , w ∈ Ω0

for some choice of anti-holomorphic functionsK(1), . . . ,K(d) : Ω0 →M ,

the vectors K(i)w , 1 ≤ i ≤ d , are linearly independent in M for w in Ω0

the vectors K(i)w0 | 1 ≤ i ≤ d are uniquely determined by these

generators g01 , . . . , g

0d ,

the linear span of the set of vectors K(i)w0 | 1 ≤ i ≤ d in M is

independent of the generators g01 , . . . , g

0d , and

the vectors K(i)w0 , 1 ≤ i ≤ d , are eigenvectors for the adjoint of the

action of C[z] on the Hilbert module M at w0 .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

The decomposition theorem

Theorem. Suppose g0i , 1 ≤ i ≤ d, be a minimal set of generators for

the stalk SMw0 . Then there exists a open neighborhood Ω0 of w0 such

that

K(·, w) := Kw = g01(w)K(1)

w + · · ·+ g0d(w)K(d)

w , w ∈ Ω0

for some choice of anti-holomorphic functionsK(1), . . . ,K(d) : Ω0 →M ,

the vectors K(i)w , 1 ≤ i ≤ d , are linearly independent in M for w in Ω0

the vectors K(i)w0 | 1 ≤ i ≤ d are uniquely determined by these

generators g01 , . . . , g

0d ,

the linear span of the set of vectors K(i)w0 | 1 ≤ i ≤ d in M is

independent of the generators g01 , . . . , g

0d , and

the vectors K(i)w0 , 1 ≤ i ≤ d , are eigenvectors for the adjoint of the

action of C[z] on the Hilbert module M at w0 .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

The decomposition theorem

Theorem. Suppose g0i , 1 ≤ i ≤ d, be a minimal set of generators for

the stalk SMw0 . Then there exists a open neighborhood Ω0 of w0 such

that

K(·, w) := Kw = g01(w)K(1)

w + · · ·+ g0d(w)K(d)

w , w ∈ Ω0

for some choice of anti-holomorphic functionsK(1), . . . ,K(d) : Ω0 →M ,

the vectors K(i)w , 1 ≤ i ≤ d , are linearly independent in M for w in Ω0

the vectors K(i)w0 | 1 ≤ i ≤ d are uniquely determined by these

generators g01 , . . . , g

0d ,

the linear span of the set of vectors K(i)w0 | 1 ≤ i ≤ d in M is

independent of the generators g01 , . . . , g

0d , and

the vectors K(i)w0 , 1 ≤ i ≤ d , are eigenvectors for the adjoint of the

action of C[z] on the Hilbert module M at w0 .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

Key ingredients of the proof of the Theorem

Every function in a submodule of Ow0 decomposes in terms of itsgenerator on a small neighbourhood of w0 , with coefficientssatisfying some norm bounds in a even smaller compactneighbourhood of 0 .

Ow0 is a local ring and the Nakayama’s lemma

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

Key ingredients of the proof of the Theorem

Every function in a submodule of Ow0 decomposes in terms of itsgenerator on a small neighbourhood of w0 , with coefficientssatisfying some norm bounds in a even smaller compactneighbourhood of 0 .

Ow0 is a local ring and the Nakayama’s lemma

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

The isomorphism of Modules

For any two Hilbert module M1 and M2 in the class B1(Ω) andL : M1 →M2 a module map between them, letSL : SM1(V )→ SM2(V ) be the map defined by

SL

n∑i=1

fi|V gi :=n∑i=1

Lfi|V gi, for fi ∈M1, gi ∈ O(V ), n ∈ N.

The map SL is well defined: if∑ni=1 fi|V gi =

∑ni=1 fi|V gi, then∑n

i=1 Lfi|V gi =∑ni=1 Lfi|V gi.

Suppose M1 is isomorphic to M2 via a unitary module map L. Now,it is easy to verify that (SL)−1 = SL

∗. It then follows that SM1 is

isomorphic, as a sheaf of modules over OΩ, to SM2 via the map SL.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

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The decomposition theoremRigidity TheoremAn inequality

The isomorphism of Modules

For any two Hilbert module M1 and M2 in the class B1(Ω) andL : M1 →M2 a module map between them, letSL : SM1(V )→ SM2(V ) be the map defined by

SL

n∑i=1

fi|V gi :=n∑i=1

Lfi|V gi, for fi ∈M1, gi ∈ O(V ), n ∈ N.

The map SL is well defined: if∑ni=1 fi|V gi =

∑ni=1 fi|V gi, then∑n

i=1 Lfi|V gi =∑ni=1 Lfi|V gi.

Suppose M1 is isomorphic to M2 via a unitary module map L. Now,it is easy to verify that (SL)−1 = SL

∗. It then follows that SM1 is

isomorphic, as a sheaf of modules over OΩ, to SM2 via the map SL.

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Now, equality

Let Mi ∈ B1(Ω \X), i = 1, 2 w0 ∈ X, for some closed analytic subsetX ⊂ Ω with dimX ≤ m− 2 . For w0 ∈ X the stalks are not justisomorphic but equal:

SM1w0 =

n∑i=1

higi : gi ∈M1, hi ∈ Ow0 , 1 ≤ i ≤ n, n ∈ N

= n∑i=1

hiφfi : fi ∈M2, hi ∈ Ow0 , 1 ≤ i ≤ n, n ∈ N

= n∑i=1

hifi : fi ∈M2, hi ∈ Ow0 , 1 ≤ i ≤ n, n ∈ N = SM2w0 .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

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The decomposition theoremRigidity TheoremAn inequality

Now, equality

Let Mi ∈ B1(Ω \X), i = 1, 2 w0 ∈ X, for some closed analytic subsetX ⊂ Ω with dimX ≤ m− 2 . For w0 ∈ X the stalks are not justisomorphic but equal:

SM1w0 =

n∑i=1

higi : gi ∈M1, hi ∈ Ow0 , 1 ≤ i ≤ n, n ∈ N

= n∑i=1

hiφfi : fi ∈M2, hi ∈ Ow0 , 1 ≤ i ≤ n, n ∈ N

= n∑i=1

hifi : fi ∈M2, hi ∈ Ow0 , 1 ≤ i ≤ n, n ∈ N = SM2w0 .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

Now, equality

Let Mi ∈ B1(Ω \X), i = 1, 2 w0 ∈ X, for some closed analytic subsetX ⊂ Ω with dimX ≤ m− 2 . For w0 ∈ X the stalks are not justisomorphic but equal:

SM1w0 =

n∑i=1

higi : gi ∈M1, hi ∈ Ow0 , 1 ≤ i ≤ n, n ∈ N

= n∑i=1

hiφfi : fi ∈M2, hi ∈ Ow0 , 1 ≤ i ≤ n, n ∈ N

= n∑i=1

hifi : fi ∈M2, hi ∈ Ow0 , 1 ≤ i ≤ n, n ∈ N = SM2w0 .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

The decomposition theoremRigidity TheoremAn inequality

Applications

Theorem. Let M ⊆ O(Ω) and M ⊆ O(Ω) be two Hilbert modules ofthe form [I] and [I], respectively, where I, I are polynomial ideals.Assume that M, M are in B1(Ω) and that the dimension of the zeroset of these modules is at most m− 2. Furthermore, also assume thatevery algebraic component of V (I) and V (I) intersects Ω. If M

and M are equivalent, then I = I.

Question is if we consider the closure of the ideal I in two differentHilbert modules M and M , whether they can be equivalent or not.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

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The decomposition theoremRigidity TheoremAn inequality

Applications

Theorem. Let M ⊆ O(Ω) and M ⊆ O(Ω) be two Hilbert modules ofthe form [I] and [I], respectively, where I, I are polynomial ideals.Assume that M, M are in B1(Ω) and that the dimension of the zeroset of these modules is at most m− 2. Furthermore, also assume thatevery algebraic component of V (I) and V (I) intersects Ω. If M

and M are equivalent, then I = I.

Question is if we consider the closure of the ideal I in two differentHilbert modules M and M , whether they can be equivalent or not.

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The decomposition theoremRigidity TheoremAn inequality

In terms of no. of minimal generators

From the decomposition Theorem, it follows that

dim kerD(M−w0)∗ ≥ dim SMw0/mw0S

Mw0 .

Proposition. Let M = [I] be a submodule of an analytic Hilbertmodule over C[z] , where I is an ideal in the polynomial ring C[z] .Then

dim SMw0/mw0S

Mw0 = ]minimal set of generators for S

Mw0 = dim kerD(M−w0)∗ .

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The decomposition theoremRigidity TheoremAn inequality

In terms of no. of minimal generators

From the decomposition Theorem, it follows that

dim kerD(M−w0)∗ ≥ dim SMw0/mw0S

Mw0 .

Proposition. Let M = [I] be a submodule of an analytic Hilbertmodule over C[z] , where I is an ideal in the polynomial ring C[z] .Then

dim SMw0/mw0S

Mw0 = ]minimal set of generators for S

Mw0 = dim kerD(M−w0)∗ .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

Back to H20 (D2)

In the example of the module H20 (D2) , we have

SH2

0 (D2)w =

Ow ifw 6= (0, 0)

m(0,0)O(0,0) if w = (0, 0).

While the germs of holomorphic function Ow at w ∈ D2 is singlygenarated (even if w = (0, 0) ), the ideal m(0,0)O(0,0) ⊆ O(0,0) is 2 -generated.

Thus the number of generators match the dimension of the jointeigenspace, in this case.

The reproducing kernel KH20 (D2)(z, w) is easy to compute:

KH2(D2)(z, w)− 1 =z1w1 + z2w2 − z1z2w1w2

(1− z1w1)(1− z2w2).

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

Back to H20 (D2)

In the example of the module H20 (D2) , we have

SH2

0 (D2)w =

Ow ifw 6= (0, 0)

m(0,0)O(0,0) if w = (0, 0).

While the germs of holomorphic function Ow at w ∈ D2 is singlygenarated (even if w = (0, 0) ), the ideal m(0,0)O(0,0) ⊆ O(0,0) is 2 -generated.

Thus the number of generators match the dimension of the jointeigenspace, in this case.

The reproducing kernel KH20 (D2)(z, w) is easy to compute:

KH2(D2)(z, w)− 1 =z1w1 + z2w2 − z1z2w1w2

(1− z1w1)(1− z2w2).

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

Back to H20 (D2)

In the example of the module H20 (D2) , we have

SH2

0 (D2)w =

Ow ifw 6= (0, 0)

m(0,0)O(0,0) if w = (0, 0).

While the germs of holomorphic function Ow at w ∈ D2 is singlygenarated (even if w = (0, 0) ), the ideal m(0,0)O(0,0) ⊆ O(0,0) is 2 -generated.

Thus the number of generators match the dimension of the jointeigenspace, in this case.

The reproducing kernel KH20 (D2)(z, w) is easy to compute:

KH2(D2)(z, w)− 1 =z1w1 + z2w2 − z1z2w1w2

(1− z1w1)(1− z2w2).

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

Back to H20 (D2)

In the example of the module H20 (D2) , we have

SH2

0 (D2)w =

Ow ifw 6= (0, 0)

m(0,0)O(0,0) if w = (0, 0).

While the germs of holomorphic function Ow at w ∈ D2 is singlygenarated (even if w = (0, 0) ), the ideal m(0,0)O(0,0) ⊆ O(0,0) is 2 -generated.

Thus the number of generators match the dimension of the jointeigenspace, in this case.

The reproducing kernel KH20 (D2)(z, w) is easy to compute:

KH2(D2)(z, w)− 1 =z1w1 + z2w2 − z1z2w1w2

(1− z1w1)(1− z2w2).

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MotivationThe sheaf model

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ExampleGeneral setupInvariants

H20 (D2) example contd.

How do we find the unique pair of vectors K(1)0 and K

(2)0 ?

set w1θ1 = w2 for w1 6= 0 , and take the limit:

lim(w1,w2)→0

KH20 (D2)(z, w)

w1= K

(1)0 (z) + θ1K

(2)0 (z) = z1 + θ1z2

to obtain K(1)0 and K

(2)0 by the uniqueness in the Decomposition

Theorem. Similarly, for w2θ2 = w1 with w2 6= 0, we have

lim(w1,w2)→0

KH20 (D2)(z, w)

w2= K

(2)0 (z) + θ2K

(1)0 (z) = z2 + θ2z1.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

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ExampleGeneral setupInvariants

H20 (D2) example contd.

How do we find the unique pair of vectors K(1)0 and K

(2)0 ?

set w1θ1 = w2 for w1 6= 0 , and take the limit:

lim(w1,w2)→0

KH20 (D2)(z, w)

w1= K

(1)0 (z) + θ1K

(2)0 (z) = z1 + θ1z2

to obtain K(1)0 and K

(2)0 by the uniqueness in the Decomposition

Theorem. Similarly, for w2θ2 = w1 with w2 6= 0, we have

lim(w1,w2)→0

KH20 (D2)(z, w)

w2= K

(2)0 (z) + θ2K

(1)0 (z) = z2 + θ2z1.

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ExampleGeneral setupInvariants

Blowing up the origin

Thus we have a Hermitian line bundle on the complex projectivespace P1 given by the frame θ1 7→ z1 + θ1z2 and θ2 7→ z2 + θ2z1.

The curvature of this line bundle is then an invariant for the Hilbertmodule H2

0 (D2). This curvature is easily calculated and is given by theformula K(θ) = (1 + |θ|2)−2.

The decomposition theorem yields similar results in many otherexamples.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

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ExampleGeneral setupInvariants

Blowing up the origin

Thus we have a Hermitian line bundle on the complex projectivespace P1 given by the frame θ1 7→ z1 + θ1z2 and θ2 7→ z2 + θ2z1.

The curvature of this line bundle is then an invariant for the Hilbertmodule H2

0 (D2). This curvature is easily calculated and is given by theformula K(θ) = (1 + |θ|2)−2.

The decomposition theorem yields similar results in many otherexamples.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

Blowing up the origin

Thus we have a Hermitian line bundle on the complex projectivespace P1 given by the frame θ1 7→ z1 + θ1z2 and θ2 7→ z2 + θ2z1.

The curvature of this line bundle is then an invariant for the Hilbertmodule H2

0 (D2). This curvature is easily calculated and is given by theformula K(θ) = (1 + |θ|2)−2.

The decomposition theorem yields similar results in many otherexamples.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

In general ...

Let M be a Hilbert module in B1(Ω), which is the closure, in M, ofsome polynomial ideal I. Let K denote the correspondingreproducing kernel. Let w0 ∈ V (M). Set

t = dim SMw0/mw0S

Mw0 = dim∩mj=1 ker(Mj − w0j)

∗

By the Decomposition Theorem, there exists a minimal set ofgenerators g1, · · · , gt of S

M10 and a r > 0 such that

K(·, w) =

t∑i=1

gj(w)K(j)(·, w) for all w ∈ ∆(w0; r)

for some choice of anti-holomorphic functionsK(1), . . . ,K(t) : ∆(w0; r)→M.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

In general ...

Let M be a Hilbert module in B1(Ω), which is the closure, in M, ofsome polynomial ideal I. Let K denote the correspondingreproducing kernel. Let w0 ∈ V (M). Set

t = dim SMw0/mw0S

Mw0 = dim∩mj=1 ker(Mj − w0j)

∗

By the Decomposition Theorem, there exists a minimal set ofgenerators g1, · · · , gt of S

M10 and a r > 0 such that

K(·, w) =

t∑i=1

gj(w)K(j)(·, w) for all w ∈ ∆(w0; r)

for some choice of anti-holomorphic functionsK(1), . . . ,K(t) : ∆(w0; r)→M.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

In general ...

Let M be a Hilbert module in B1(Ω), which is the closure, in M, ofsome polynomial ideal I. Let K denote the correspondingreproducing kernel. Let w0 ∈ V (M). Set

t = dim SMw0/mw0S

Mw0 = dim∩mj=1 ker(Mj − w0j)

∗

By the Decomposition Theorem, there exists a minimal set ofgenerators g1, · · · , gt of S

M10 and a r > 0 such that

K(·, w) =

t∑i=1

gj(w)K(j)(·, w) for all w ∈ ∆(w0; r)

for some choice of anti-holomorphic functionsK(1), . . . ,K(t) : ∆(w0; r)→M.

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In general ...

Assume that Z := V (g1, . . . , gt) ∩ Ω be a singularity free analytic subsetof Cm of codimension t.

∆(w0; r) = (w, π(u)) ∈ ∆(w0; r)× Pt−1 : uigj(w)− ujgi(w) = 0, 1 ≤ i, j ≤ t,

where the map π : Ct \ 0 → Pt−1 is given by π(u) = (u1 : . . . : ut).

The space ∆(w0; r) is the monoidal transformation with center Z.

Consider the map p := pr1 : ∆(w0; r)→ ∆(w0; r) given by(w, π(z)) 7→ w. For w ∈ Z, we have p−1(w) = w × Pt−1. This map isholomorphic and proper.

Actually p : ∆(w0; r) \ p−1(Z)→ ∆(w0; r) \ Z is biholomorphic withp−1 : w 7→ (w, (g1(w) : . . . : gt(w))). The set E(M) := p−1(Z) which isZ × Pt−1 is called the exceptional set.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

In general ...

Assume that Z := V (g1, . . . , gt) ∩ Ω be a singularity free analytic subsetof Cm of codimension t.

∆(w0; r) = (w, π(u)) ∈ ∆(w0; r)× Pt−1 : uigj(w)− ujgi(w) = 0, 1 ≤ i, j ≤ t,

where the map π : Ct \ 0 → Pt−1 is given by π(u) = (u1 : . . . : ut).

The space ∆(w0; r) is the monoidal transformation with center Z.

Consider the map p := pr1 : ∆(w0; r)→ ∆(w0; r) given by(w, π(z)) 7→ w. For w ∈ Z, we have p−1(w) = w × Pt−1. This map isholomorphic and proper.

Actually p : ∆(w0; r) \ p−1(Z)→ ∆(w0; r) \ Z is biholomorphic withp−1 : w 7→ (w, (g1(w) : . . . : gt(w))). The set E(M) := p−1(Z) which isZ × Pt−1 is called the exceptional set.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

In general ...

Assume that Z := V (g1, . . . , gt) ∩ Ω be a singularity free analytic subsetof Cm of codimension t.

∆(w0; r) = (w, π(u)) ∈ ∆(w0; r)× Pt−1 : uigj(w)− ujgi(w) = 0, 1 ≤ i, j ≤ t,

where the map π : Ct \ 0 → Pt−1 is given by π(u) = (u1 : . . . : ut).

The space ∆(w0; r) is the monoidal transformation with center Z.

Consider the map p := pr1 : ∆(w0; r)→ ∆(w0; r) given by(w, π(z)) 7→ w. For w ∈ Z, we have p−1(w) = w × Pt−1. This map isholomorphic and proper.

Actually p : ∆(w0; r) \ p−1(Z)→ ∆(w0; r) \ Z is biholomorphic withp−1 : w 7→ (w, (g1(w) : . . . : gt(w))). The set E(M) := p−1(Z) which isZ × Pt−1 is called the exceptional set.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

In general ...

Assume that Z := V (g1, . . . , gt) ∩ Ω be a singularity free analytic subsetof Cm of codimension t.

∆(w0; r) = (w, π(u)) ∈ ∆(w0; r)× Pt−1 : uigj(w)− ujgi(w) = 0, 1 ≤ i, j ≤ t,

where the map π : Ct \ 0 → Pt−1 is given by π(u) = (u1 : . . . : ut).

The space ∆(w0; r) is the monoidal transformation with center Z.

Consider the map p := pr1 : ∆(w0; r)→ ∆(w0; r) given by(w, π(z)) 7→ w. For w ∈ Z, we have p−1(w) = w × Pt−1. This map isholomorphic and proper.

Actually p : ∆(w0; r) \ p−1(Z)→ ∆(w0; r) \ Z is biholomorphic withp−1 : w 7→ (w, (g1(w) : . . . : gt(w))). The set E(M) := p−1(Z) which isZ × Pt−1 is called the exceptional set.

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MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

New vector bundles

Consider the open set U1 = (∆(w0; r)× u1 6= 0) ∩ ∆(w0; r). Letuju1

= θ1j , 2 ≤ j ≤ t. On this chart gj(w) = θ1

j gj(w). From thedecomposition for the K(·, w), we have

K(·, w) = g1(w)K(1)(·, w) +

t∑j=2

θ1jK

(j)(·, w).

This decomposition then yields a section on the chart U1, of the linebundle on the blow-up space ∆(w0; r):

s1(w, θ) = K(1)(·, w) +

t∑j=2

θ1jK

(j)(·, w).

The vectors K(j)(·, w) are not uniquely determined. However, thereexists a canonical choice of these vectors starting from a basis,v1, . . . , vt, of the joint kernel ∩ni=1 ker(Mj − wj)∗:

K(·, w) =

t∑j=1

gj(w)P (w, w0)vj , w ∈ ∆(w0; r)

for some r > 0 and generators g1, . . . , gt of the stalk SMw0 .

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MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

New vector bundles

Consider the open set U1 = (∆(w0; r)× u1 6= 0) ∩ ∆(w0; r). Letuju1

= θ1j , 2 ≤ j ≤ t. On this chart gj(w) = θ1

j gj(w). From thedecomposition for the K(·, w), we have

K(·, w) = g1(w)K(1)(·, w) +

t∑j=2

θ1jK

(j)(·, w).

This decomposition then yields a section on the chart U1, of the linebundle on the blow-up space ∆(w0; r):

s1(w, θ) = K(1)(·, w) +

t∑j=2

θ1jK

(j)(·, w).

The vectors K(j)(·, w) are not uniquely determined. However, thereexists a canonical choice of these vectors starting from a basis,v1, . . . , vt, of the joint kernel ∩ni=1 ker(Mj − wj)∗:

K(·, w) =t∑

j=1

gj(w)P (w, w0)vj , w ∈ ∆(w0; r)

for some r > 0 and generators g1, . . . , gt of the stalk SMw0 .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

New vector bundles

Consider the open set U1 = (∆(w0; r)× u1 6= 0) ∩ ∆(w0; r). Letuju1

= θ1j , 2 ≤ j ≤ t. On this chart gj(w) = θ1

j gj(w). From thedecomposition for the K(·, w), we have

K(·, w) = g1(w)K(1)(·, w) +

t∑j=2

θ1jK

(j)(·, w).

This decomposition then yields a section on the chart U1, of the linebundle on the blow-up space ∆(w0; r):

s1(w, θ) = K(1)(·, w) +

t∑j=2

θ1jK

(j)(·, w).

The vectors K(j)(·, w) are not uniquely determined. However, thereexists a canonical choice of these vectors starting from a basis,v1, . . . , vt, of the joint kernel ∩ni=1 ker(Mj − wj)∗:

K(·, w) =t∑

j=1

gj(w)P (w, w0)vj , w ∈ ∆(w0; r)

for some r > 0 and generators g1, . . . , gt of the stalk SMw0 .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

Let L(M) be the line bundle on the blow-up space ∆(w0; r)determined by the section (w, θ) 7→ s1(w, θ), where

s1(w, θ) = P (w, w0)v1 +t∑j=2

θ1jP (w, w0)vj , (w, θ) ∈ U1.

Theorem. Let M ⊆ O(Ω) and M ⊆ O(Ω) be two Hilbert modules ofthe form [I] and [I] , respectively, where I, I are polynomial ideals.Assume that M , M are in B1(Ω) and that the dimension of the zeroset of these modules is at most m− 2 . Then M and M are equivalentif and only if the line bundles L(M) and L(M) are equivalent asHermitian holomorphic line bundle on ∆(w0; r)∗ .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

Let L(M) be the line bundle on the blow-up space ∆(w0; r)determined by the section (w, θ) 7→ s1(w, θ), where

s1(w, θ) = P (w, w0)v1 +t∑j=2

θ1jP (w, w0)vj , (w, θ) ∈ U1.

Theorem. Let M ⊆ O(Ω) and M ⊆ O(Ω) be two Hilbert modules ofthe form [I] and [I] , respectively, where I, I are polynomial ideals.Assume that M , M are in B1(Ω) and that the dimension of the zeroset of these modules is at most m− 2 . Then M and M are equivalentif and only if the line bundles L(M) and L(M) are equivalent asHermitian holomorphic line bundle on ∆(w0; r)∗ .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

Restriction to the exceptional set

In general, Z need not be a complex manifold. However, therestriction of s1 to p−1(w0) for w0 ∈ Z determines a holomorphic linebundle L0(M) on p−1(w0)∗ which is the set(w0, π(u)) : (w0, π(u)) ∈ p−1(w0) . Thus s1 = s1(w, θ)|w0×ui 6=0 isgiven by the formula

s1(θ) = K(1)(·, w0) +

t∑j=2

θ1jK

(j)(·, w0).

Since the vectors K(j)(·, w0), 1 ≤ j ≤ t are uniquely determined bythe generators g1, . . . , gt, s1 is well defined.

Theorem. Let M ⊆ O(Ω) and M ⊆ O(Ω) be two Hilbert modules ofthe form [I] and [I] , respectively and I, I ⊆ C[z] . Assume that M, M

are in B1(Ω) and that the dimension of the zero set of these modulesis at most m− 2. If the modules M and M are equivalent, then thecorresponding bundles L0(M) and L0(M) they determine on theprojective space p−1(w0)∗ for w0 ∈ Z, are equivalent as Hermitianholomorphic line bundle.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

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ExampleGeneral setupInvariants

Restriction to the exceptional set

In general, Z need not be a complex manifold. However, therestriction of s1 to p−1(w0) for w0 ∈ Z determines a holomorphic linebundle L0(M) on p−1(w0)∗ which is the set(w0, π(u)) : (w0, π(u)) ∈ p−1(w0) . Thus s1 = s1(w, θ)|w0×ui 6=0 isgiven by the formula

s1(θ) = K(1)(·, w0) +

t∑j=2

θ1jK

(j)(·, w0).

Since the vectors K(j)(·, w0), 1 ≤ j ≤ t are uniquely determined bythe generators g1, . . . , gt, s1 is well defined.

Theorem. Let M ⊆ O(Ω) and M ⊆ O(Ω) be two Hilbert modules ofthe form [I] and [I] , respectively and I, I ⊆ C[z] . Assume that M, M

are in B1(Ω) and that the dimension of the zero set of these modulesis at most m− 2. If the modules M and M are equivalent, then thecorresponding bundles L0(M) and L0(M) they determine on theprojective space p−1(w0)∗ for w0 ∈ Z, are equivalent as Hermitianholomorphic line bundle.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

Restriction to the exceptional set

In general, Z need not be a complex manifold. However, therestriction of s1 to p−1(w0) for w0 ∈ Z determines a holomorphic linebundle L0(M) on p−1(w0)∗ which is the set(w0, π(u)) : (w0, π(u)) ∈ p−1(w0) . Thus s1 = s1(w, θ)|w0×ui 6=0 isgiven by the formula

s1(θ) = K(1)(·, w0) +

t∑j=2

θ1jK

(j)(·, w0).

Since the vectors K(j)(·, w0), 1 ≤ j ≤ t are uniquely determined bythe generators g1, . . . , gt, s1 is well defined.

Theorem. Let M ⊆ O(Ω) and M ⊆ O(Ω) be two Hilbert modules ofthe form [I] and [I] , respectively and I, I ⊆ C[z] . Assume that M, M

are in B1(Ω) and that the dimension of the zero set of these modulesis at most m− 2. If the modules M and M are equivalent, then thecorresponding bundles L0(M) and L0(M) they determine on theprojective space p−1(w0)∗ for w0 ∈ Z, are equivalent as Hermitianholomorphic line bundle.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

Examples

Let B2 be the unit ball in C2. For −1 < α, β, θ < +∞, let L2α,β,θ(B2) be

the Hilbert space of functions on B2 satisfying

‖ f ‖2α,β,θ=∫

B2|f(z)|2dµ(z1, z2) < +∞,

dµ(z1, z2) = (α+ β + θ + 2)|z2|2θ(1− |z1|2 − |z2|2)α(1− |z2|2)βdA(z1, z2)and dA(z1, z2) = dA(z1)dA(z2).

The weighted Bergman space A2α,β,θ(B2) is the subspace of

L2α,β,θ(B2) consisting of the holomorphic functions on B2. The Hilbert

space A2α,β,θ(B2) is non-trivial if we assume that the parameters α, β, θ

satisfy the additional condition: α+ β + θ + 2 > 0.

Proposition. Suppose I is an ideal in C[z1, z2] with V (I) = 0. Thenthe Hilbert modules [I]A2

α,β,θ(B2) and [I]A2

α′,β′,θ′ (B2) are unitarily

equivalent if and only if α = α′, β = β′ and θ = θ′.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

ExampleGeneral setupInvariants

Examples

Let B2 be the unit ball in C2. For −1 < α, β, θ < +∞, let L2α,β,θ(B2) be

the Hilbert space of functions on B2 satisfying

‖ f ‖2α,β,θ=∫

B2|f(z)|2dµ(z1, z2) < +∞,

dµ(z1, z2) = (α+ β + θ + 2)|z2|2θ(1− |z1|2 − |z2|2)α(1− |z2|2)βdA(z1, z2)and dA(z1, z2) = dA(z1)dA(z2).

The weighted Bergman space A2α,β,θ(B2) is the subspace of

L2α,β,θ(B2) consisting of the holomorphic functions on B2. The Hilbert

space A2α,β,θ(B2) is non-trivial if we assume that the parameters α, β, θ

satisfy the additional condition: α+ β + θ + 2 > 0.

Proposition. Suppose I is an ideal in C[z1, z2] with V (I) = 0. Thenthe Hilbert modules [I]A2

α,β,θ(B2) and [I]A2

α′,β′,θ′ (B2) are unitarily

equivalent if and only if α = α′, β = β′ and θ = θ′.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

Couple of questionsDescription of basis vectors

Examples

Let I0 be the polynomial ideal 〈z1, z2〉 and [I0] denotes its closure inH2(D2) .

Note that ∂1K[I0](z, w)|w=0 and ∂2K[I0](z, w)|w=0 forms a basis of thejoint kernel at 0 .

Next let I1 be the polynomial ideal 〈z1 − z2, z22〉 and [I1] denotes its

closure in H2(D2) .

Note that ∂22K[I1](z, w)|w=0 doesn’t even belong to the joint kernel at

0 .

A set of two linearly independent vectors which span it arep1(∂1, ∂2)K[I1](z, w)|w=0 and p2(∂1, ∂2)K[I1](z, w)|w=0 , wherep1 = z1 − z2 and p2 = (z1 + z2)2 .

Unlike the first example, the two polynomials p1, p2 are not thegenerators for the ideal I1 that were given at the start, never the less,they are easily seen to be a set of generators for the ideal I1 as well.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

Couple of questionsDescription of basis vectors

Examples

Let I0 be the polynomial ideal 〈z1, z2〉 and [I0] denotes its closure inH2(D2) .

Note that ∂1K[I0](z, w)|w=0 and ∂2K[I0](z, w)|w=0 forms a basis of thejoint kernel at 0 .

Next let I1 be the polynomial ideal 〈z1 − z2, z22〉 and [I1] denotes its

closure in H2(D2) .

Note that ∂22K[I1](z, w)|w=0 doesn’t even belong to the joint kernel at

0 .

A set of two linearly independent vectors which span it arep1(∂1, ∂2)K[I1](z, w)|w=0 and p2(∂1, ∂2)K[I1](z, w)|w=0 , wherep1 = z1 − z2 and p2 = (z1 + z2)2 .

Unlike the first example, the two polynomials p1, p2 are not thegenerators for the ideal I1 that were given at the start, never the less,they are easily seen to be a set of generators for the ideal I1 as well.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

Couple of questionsDescription of basis vectors

Examples

Let I0 be the polynomial ideal 〈z1, z2〉 and [I0] denotes its closure inH2(D2) .

Note that ∂1K[I0](z, w)|w=0 and ∂2K[I0](z, w)|w=0 forms a basis of thejoint kernel at 0 .

Next let I1 be the polynomial ideal 〈z1 − z2, z22〉 and [I1] denotes its

closure in H2(D2) .

Note that ∂22K[I1](z, w)|w=0 doesn’t even belong to the joint kernel at

0 .

A set of two linearly independent vectors which span it arep1(∂1, ∂2)K[I1](z, w)|w=0 and p2(∂1, ∂2)K[I1](z, w)|w=0 , wherep1 = z1 − z2 and p2 = (z1 + z2)2 .

Unlike the first example, the two polynomials p1, p2 are not thegenerators for the ideal I1 that were given at the start, never the less,they are easily seen to be a set of generators for the ideal I1 as well.

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

Couple of questionsDescription of basis vectors

Question

Let M = [I] ∈ B1(Ω) be a Hilbert module, where I ⊆M is apolynomial ideal. Assume without loss of generality that 0 ∈ V (I) . Weask

1 if there exists a set of polynomials p1, . . . , pt such that

pi(∂∂w1

, . . . , ∂∂wm

)K[I](z, w)|w=0, i = 1, . . . , t,

forms a basis of the joint kernel of [I] ;2 what conditions, if any, will ensure that the polynomials p1, . . . , pt ,

as above, is a generating set for I ?

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

Couple of questionsDescription of basis vectors

A description

Let M = [I] ∈ B1(Ω) .

The characteristic space of I at w ∈ Ω is

Vw(I) := q ∈ C[z] : q(D)p∣∣w

= 0, p ∈ I.

Also letVw(I) := q ∈ C[z] :

∂q

∂zi∈ Vw(I), 1 ≤ i ≤ m.

Lemma. Fix w0 ∈ Ω and polynomials q1, . . . , qt . Let I be apolynomial ideal and K be the reproducing kernel corresponding theHilbert module [I] , which is assumed to be in B1(Ω) . Then the vectors

q1(D)K(·, w)|w=w0 , . . . , qt(D)K(·, w)|w=w0

form a basis of the joint kernel at w0 of the adjoint of themultiplication operator if and only if the classes [q∗1 ], . . . , [q∗t ] form abasis of Vw0(I)/Vw0(I) .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

Couple of questionsDescription of basis vectors

A description

Let M = [I] ∈ B1(Ω) .

The characteristic space of I at w ∈ Ω is

Vw(I) := q ∈ C[z] : q(D)p∣∣w

= 0, p ∈ I.

Also letVw(I) := q ∈ C[z] :

∂q

∂zi∈ Vw(I), 1 ≤ i ≤ m.

Lemma. Fix w0 ∈ Ω and polynomials q1, . . . , qt . Let I be apolynomial ideal and K be the reproducing kernel corresponding theHilbert module [I] , which is assumed to be in B1(Ω) . Then the vectors

q1(D)K(·, w)|w=w0 , . . . , qt(D)K(·, w)|w=w0

form a basis of the joint kernel at w0 of the adjoint of themultiplication operator if and only if the classes [q∗1 ], . . . , [q∗t ] form abasis of Vw0(I)/Vw0(I) .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

Couple of questionsDescription of basis vectors

In terms of generators

Consider the ideal I generated by the polynomialsz1 + z2 + z2

1 , z32 − z2

1 . The joint kernel at 0 , in this case is spanned by theindependent vectors p(D)K(·, w)|w=0, q(D)K(·, w)|w=0 , wherep = z1 + z2 and q = (z1 − z2)2 . A little bit more calculation shows thatthere does not exist a set of minimal generators p, q for I such thatp(D)K(·, w)|w=0, q(D)K(·, w)|w=0 is a basis of the joint kernel.

Theorem. Let I ⊂ C[z] be a homogeneous ideal with rankC[z]I = v .Let K be the reproducing kernel corresponding to the Hilbert module[I] , which is assumed to be in B1(Ω) . Then there exists a set ofgenerators q1, ..., qv for the ideal I such that the setqi(D)K(·, w)|w=0 : 1 ≤ i ≤ v is a basis for ∩mj=1 kerM∗j .

Proof of the Theorem above is based on realizing the coefficient matrixof certain system of linear equations as a sum of the Grammian ofcertain vectors with respect to Fock inner product〈p, q〉 = q∗(D)p|0, q∗(z) = q(z) for p, q ∈ C[z] .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

Couple of questionsDescription of basis vectors

In terms of generators

Consider the ideal I generated by the polynomialsz1 + z2 + z2

1 , z32 − z2

1 . The joint kernel at 0 , in this case is spanned by theindependent vectors p(D)K(·, w)|w=0, q(D)K(·, w)|w=0 , wherep = z1 + z2 and q = (z1 − z2)2 . A little bit more calculation shows thatthere does not exist a set of minimal generators p, q for I such thatp(D)K(·, w)|w=0, q(D)K(·, w)|w=0 is a basis of the joint kernel.

Theorem. Let I ⊂ C[z] be a homogeneous ideal with rankC[z]I = v .Let K be the reproducing kernel corresponding to the Hilbert module[I] , which is assumed to be in B1(Ω) . Then there exists a set ofgenerators q1, ..., qv for the ideal I such that the setqi(D)K(·, w)|w=0 : 1 ≤ i ≤ v is a basis for ∩mj=1 kerM∗j .

Proof of the Theorem above is based on realizing the coefficient matrixof certain system of linear equations as a sum of the Grammian ofcertain vectors with respect to Fock inner product〈p, q〉 = q∗(D)p|0, q∗(z) = q(z) for p, q ∈ C[z] .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

Couple of questionsDescription of basis vectors

In terms of generators

Consider the ideal I generated by the polynomialsz1 + z2 + z2

1 , z32 − z2

1 . The joint kernel at 0 , in this case is spanned by theindependent vectors p(D)K(·, w)|w=0, q(D)K(·, w)|w=0 , wherep = z1 + z2 and q = (z1 − z2)2 . A little bit more calculation shows thatthere does not exist a set of minimal generators p, q for I such thatp(D)K(·, w)|w=0, q(D)K(·, w)|w=0 is a basis of the joint kernel.

Theorem. Let I ⊂ C[z] be a homogeneous ideal with rankC[z]I = v .Let K be the reproducing kernel corresponding to the Hilbert module[I] , which is assumed to be in B1(Ω) . Then there exists a set ofgenerators q1, ..., qv for the ideal I such that the setqi(D)K(·, w)|w=0 : 1 ≤ i ≤ v is a basis for ∩mj=1 kerM∗j .

Proof of the Theorem above is based on realizing the coefficient matrixof certain system of linear equations as a sum of the Grammian ofcertain vectors with respect to Fock inner product〈p, q〉 = q∗(D)p|0, q∗(z) = q(z) for p, q ∈ C[z] .

Shibananda Biswas , BGU Blow up technique

MotivationThe sheaf model

Blow up techniqueThe joint kernel

Couple of questionsDescription of basis vectors

Thank you!

Shibananda Biswas , BGU Blow up technique