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](https://concretenonsense.wordpress.com/2009/08/17/tangent-bundle-of-the-grassmannian/).
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 ${\rm GL}({T_X})={\Bbb C}^\times\times {\Bbb P}^1\to {\Bbb P}^1$.