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Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and $\eta^{-1}{\cal O}_E$ topological inverse image sheaf.

In the book "Penrose transform - its interaction with representation theory" by Baston and Eastwood, on page 69. there is a claim:

Sections of $\eta^{-1}{\cal O}_E$ are sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$.

There is a counterexample: Take $\eta$ to be inclusion $\{y\} \subset Z$. Then $\eta ^\ast E$ is just fiber $E_y$ over $y$, sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$ are just elements of $E_y$.

On the other hand, $\eta^{-1}{\cal O}_E$ is stalk ${\cal O}_{E,y}$, which is generally much bigger than the fiber $E_y$.

My questions are:

Are there any topological conditions on $\eta \colon Y \to Z$ that will make the claim from the book true, and how to see that?

 

Is the claim true if $\eta \colon Y \to Z$ is $G/P \to G/Q$, where $G$ is complex simply connected semisimple Lie group, the map is induced from inclusion of parabolic subgroups $P \subset Q$, and $E$ is homogeneous vector bundle?

Any help of reference is appreciated.

P.S. This question was also posted on math.stackexchange few days ago, with no answer.

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and $\eta^{-1}{\cal O}_E$ topological inverse image sheaf.

In the book "Penrose transform - its interaction with representation theory" by Baston and Eastwood, on page 69. there is a claim:

Sections of $\eta^{-1}{\cal O}_E$ are sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$.

There is a counterexample: Take $\eta$ to be inclusion $\{y\} \subset Z$. Then $\eta ^\ast E$ is just fiber $E_y$ over $y$, sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$ are just elements of $E_y$.

On the other hand, $\eta^{-1}{\cal O}_E$ is stalk ${\cal O}_{E,y}$, which is generally much bigger than the fiber $E_y$.

My questions are:

Are there any topological conditions on $\eta \colon Y \to Z$ that will make the claim from the book true, and how to see that?

Is the claim true if $\eta \colon Y \to Z$ is $G/P \to G/Q$, where $G$ is complex simply connected semisimple Lie group, the map is induced from inclusion of parabolic subgroups $P \subset Q$, and $E$ is homogeneous vector bundle?

Any help of reference is appreciated.

P.S. This question was also posted on math.stackexchange few days ago, with no answer.

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and $\eta^{-1}{\cal O}_E$ topological inverse image sheaf.

In the book "Penrose transform - its interaction with representation theory" by Baston and Eastwood, on page 69. there is a claim:

Sections of $\eta^{-1}{\cal O}_E$ are sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$.

There is a counterexample: Take $\eta$ to be inclusion $\{y\} \subset Z$. Then $\eta ^\ast E$ is just fiber $E_y$ over $y$, sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$ are just elements of $E_y$.

On the other hand, $\eta^{-1}{\cal O}_E$ is stalk ${\cal O}_{E,y}$, which is generally much bigger than the fiber $E_y$.

My questions are:

Are there any topological conditions on $\eta \colon Y \to Z$ that will make the claim from the book true, and how to see that?

Is the claim true if $\eta \colon Y \to Z$ is $G/P \to G/Q$, where $G$ is complex simply connected semisimple Lie group, the map is induced from inclusion of parabolic subgroups $P \subset Q$, and $E$ is homogeneous vector bundle?

Any help of reference is appreciated.

P.S. This question was also posted on math.stackexchange few days ago, with no answer.

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and $\eta^{-1}{\cal O}_E$ topological inverse image sheaf.

In the book "Penrose transform - its interaction with representation theory" by Baston and Eastwood, on page 69. there is a claim:

Sections of $\eta^{-1}{\cal O}_E$ are sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$.

There is a counterexample: Take $\eta$ to be inclusion $\{y\} \subset Z$. Then $\eta ^\ast E$ is just fiber $E_y$ over $y$, sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$ are just elements of $E_y$.

On the other hand, $\eta^{-1}{\cal O}_E$ is stalk ${\cal O}_{E,y}$, which is generally much bigger than the fiber $E_y$.

My questions are:

Are there any topological conditions on $\eta \colon Y \to Z$ that will make the claim from the book true, and how to see that?

Is the claim true if $\eta \colon Y \to Z$ is $G/P \to G/Q$, where $G$ is complex simply connected semisimple Lie group, the map is induced from inclusion of parabolic subgroups $P \subset Q$, and $E$ is homogeneous vector bundle?

Any help of reference is appreciated.

P.S. This question was also posted on math.stackexchange few days ago, with no answer.

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and $\eta^{-1}{\cal O}_E$ topological inverse image sheaf.

In the book "Penrose transform - its interaction with representation theory" by Baston and Eastwood, on page 69. there is a claim:

Sections of $\eta^{-1}{\cal O}_E$ are sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$.

There is a counterexample: Take $\eta$ to be inclusion $\{y\} \subset Z$. Then $\eta ^\ast E$ is just fiber $E_y$ over $y$, sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$ are just elements of $E_y$.

On the other hand, $\eta^{-1}{\cal O}_E$ is stalk ${\cal O}_{E,y}$, which is generally much bigger than the fiber $E_y$.

My questions are:

Are there any topological conditions on $\eta \colon Y \to Z$ that will make the claim from the book true, and how to see that?

Is the claim true if $\eta \colon Y \to Z$ is $G/P \to G/Q$, where $G$ is complex simply connected semisimple Lie group, the map is induced from inclusion of parabolic subgroups $P \subset Q$, and $E$ is homogeneous vector bundle?

Any reference is appreciated.

P.S. This question was also posted on math.stackexchange few days ago, with no answer.

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and $\eta^{-1}{\cal O}_E$ topological inverse image sheaf.

In the book "Penrose transform - its interaction with representation theory" by Baston and Eastwood, on page 69. there is a claim:

Sections of $\eta^{-1}{\cal O}_E$ are sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$.

There is a counterexample: Take $\eta$ to be inclusion $\{y\} \subset Z$. Then $\eta ^\ast E$ is just fiber $E_y$ over $y$, sections of bundle $\eta ^\ast E$ which are locally constant along fibers of $\eta$ are just elements of $E_y$.

On the other hand, $\eta^{-1}{\cal O}_E$ is stalk ${\cal O}_{E,y}$, which is generally much bigger than the fiber $E_y$.

My questions are:

Are there any topological conditions on $\eta \colon Y \to Z$ that will make the claim from the book true, and how to see that?

Is the claim true if $\eta \colon Y \to Z$ is $G/P \to G/Q$, where $G$ is complex simply connected semisimple Lie group, the map is induced from inclusion of parabolic subgroups $P \subset Q$, and $E$ is homogeneous vector bundle?

Any help of reference is appreciated.

P.S. This question was also posted on math.stackexchange few days ago, with no answer.