Let $G_{r}(V)$ be the Grassmannian of a complex vector space $V$ consists with subspace of codimension $r$. It is well known that $$TG_{r}(V)=Hom(S,Q)$$ where $S$ is the tautological subbundle and $Q=G_{r}\times V/S$.
I have some trouble with understanding Demailly's proof in his book "complex analytic and differential geometry". Indeed, since $Gl(V)$ acts transitively on $G_{r}(V)$, given $x\in G_{r}(V)$, let $H_{x}$ be the isotropy subgroup of $x$, then $G_{r}(V)$ is isomorphic to $Gl(V)/H_{x}=M$. So far so good.
Then we have $$
T_{x}G_{r}(V)=T_{H_{x}}M=Hom(V,V)/\{u;u(x)\subset x\}=Hom(x,V/x)=Hom(S_{x},Q_{x}).
$$
His proof ends here. My question is that how to show that this pointwise isomorphism induces a bundle isomorphism?
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J. W. Tanner
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Jiang Tianshu
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