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Let $X={\rm Gr}(k,n)$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb C}^n$. We regard $X$ as an algebraic variety over $\Bbb C$. Let ${T_X} \to X$ denote the tangent bundle on $X$. For an explicit description of ${T_X}$ see e.g. here. Consider the induced bundle ${\rm GL}({T_X})\to X$ whose fiber at $x\in X$ is the automorphism group ${\rm GL}(T_x)$ of the vector space $T_x$.

Question 1. What is the group $A={\rm Aut}_X\,{T_X}$ of regular global sections of ${\rm GL}({T_X})$ over $X$?

For any $\lambda \in{\Bbb C}^\times$ we have a global section of ${\rm GL}({T_X})$ taking value $\lambda I_x$ at $x$, where $I_x$ is the identity automorphism of $T_x$. Thus we obtain a canonical embedding ${\Bbb C}^\times\hookrightarrow A$.

Question 2. Is it true that $A={\Bbb C}^\times\,$?

 

Question 3. In particular, is the answer to Question 2 "Yes" for $X={\rm Gr}(1,n+1)={\Bbb P}^n\,$?

I know that the answer to Question 2 is "Yes" for $X={\rm Gr}(1,2)={\Bbb P}^1$. In this case ${\rm GL}({T_X})={\Bbb C}^\times\times {\Bbb P}^1\to {\Bbb P}^1$.

Let $X={\rm Gr}(k,n)$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb C}^n$. We regard $X$ as an algebraic variety over $\Bbb C$. Let ${T_X} \to X$ denote the tangent bundle on $X$. For an explicit description of ${T_X}$ see e.g. here. Consider the induced bundle ${\rm GL}({T_X})\to X$ whose fiber at $x\in X$ is the automorphism group ${\rm GL}(T_x)$ of the vector space $T_x$.

Question 1. What is the group $A={\rm Aut}_X\,{T_X}$ of regular global sections of ${\rm GL}({T_X})$ over $X$?

For any $\lambda \in{\Bbb C}^\times$ we have a global section of ${\rm GL}({T_X})$ taking value $\lambda I_x$ at $x$, where $I_x$ is the identity automorphism of $T_x$. Thus we obtain a canonical embedding ${\Bbb C}^\times\hookrightarrow A$.

Question 2. Is it true that $A={\Bbb C}^\times\,$?

 

Question 3. In particular, is the answer to Question 2 "Yes" for $X={\rm Gr}(1,n+1)={\Bbb P}^n\,$?

I know that the answer to Question 2 is "Yes" for $X={\rm Gr}(1,2)={\Bbb P}^1$. In this case ${\rm GL}({T_X})={\Bbb C}^\times\times {\Bbb P}^1\to {\Bbb P}^1$.

Let $X={\rm Gr}(k,n)$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb C}^n$. We regard $X$ as an algebraic variety over $\Bbb C$. Let ${T_X} \to X$ denote the tangent bundle on $X$. For an explicit description of ${T_X}$ see e.g. here. Consider the induced bundle ${\rm GL}({T_X})\to X$ whose fiber at $x\in X$ is the automorphism group ${\rm GL}(T_x)$ of the vector space $T_x$.

Question 1. What is the group $A={\rm Aut}_X\,{T_X}$ of regular global sections of ${\rm GL}({T_X})$ over $X$?

For any $\lambda \in{\Bbb C}^\times$ we have a global section of ${\rm GL}({T_X})$ taking value $\lambda I_x$ at $x$, where $I_x$ is the identity automorphism of $T_x$. Thus we obtain a canonical embedding ${\Bbb C}^\times\hookrightarrow A$.

Question 2. Is it true that $A={\Bbb C}^\times\,$?

Question 3. In particular, is the answer to Question 2 "Yes" for $X={\rm Gr}(1,n+1)={\Bbb P}^n\,$?

I know that the answer to Question 2 is "Yes" for $X={\rm Gr}(1,2)={\Bbb P}^1$. In this case ${\rm GL}({T_X})={\Bbb C}^\times\times {\Bbb P}^1\to {\Bbb P}^1$.

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Mikhail Borovoi
Mikhail Borovoi
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The group of global sections of the automorphism bundle of the tangent bundle on a Grassmannian

Let $X={\rm Gr}(k,n)$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb C}^n$. We regard $X$ as an algebraic variety over $\Bbb C$. Let ${T_X} \to X$ denote the tangent bundle on $X$. For an explicit description of ${T_X}$ see e.g. here. Consider the induced bundle ${\rm GL}({T_X})\to X$ whose fiber at $x\in X$ is the automorphism group ${\rm GL}(T_x)$ of the vector space $T_x$.

Question 1. What is the group $A={\rm Aut}_X\,{T_X}$ of regular global sections of ${\rm GL}({T_X})$ over $X$?

For any $\lambda \in{\Bbb C}^\times$ we have a global section of ${\rm GL}({T_X})$ taking value $\lambda I_x$ at $x$, where $I_x$ is the identity automorphism of $T_x$. Thus we obtain a canonical embedding ${\Bbb C}^\times\hookrightarrow A$.

Question 2. Is it true that $A={\Bbb C}^\times\,$?

Question 3. In particular, is the answer to Question 2 "Yes" for $X={\rm Gr}(1,n+1)={\Bbb P}^n\,$?

I know that the answer to Question 2 is "Yes" for $X={\rm Gr}(1,2)={\Bbb P}^1$. In this case ${\rm GL}({T_X})={\Bbb C}^\times\times {\Bbb P}^1\to {\Bbb P}^1$.