Normal Bundle to flag manifold/variety Let $Fl(k_1,k_2) = \{ (V_1,V_2)\in G_{k_1}(V)\times G_{k_2}(V):\text{ }V_1\subset V_2\}$ be a two step flag manifold over a $n$ - dimensional vector space $V$. I am mostly interested in the case $V =\mathbb{C}^n$. I want to construct a normal bundle to $Fl(k_1,k_2)$ in the product $G_{k_1}(V)\times G_{k_2}(V)$ using tautological bundles and I think it should be very well studied although I could not find what I was looking for in the literature.
My own construction goes as follows: Let $\gamma^{k_1}(V)$ and $\gamma^{k_2}(V)$ be the tautological bundles over $G_{k_1}(V)$ and $G_{k_2}(V)$ and fix $(V_1,V_2)\in G_{k_1}(V)\times G_{k_2}(V)$. Then there is a natural bilinear pairing $$ V_1\times (V_2)^\circ\rightarrow \mathbb{K},\quad (v,\phi)\mapsto \phi(v),$$ inducing a linear map $$s(V_1,V_2): V_1\otimes (V_2)^\circ\rightarrow\mathbb{K}.$$ By construction $s(V_1,V_2)$ is the zero map if and only if $V_1\subset V_2$. Let $\pi_i : G_{k_1}(V)\times G_{k_2}(V) \rightarrow G_{k_i}(V)$, $i=1,2$ be natural projections. Then $s$ should define a section of the bundle $$\mu := \text{Hom}(\pi_1^*(\gamma^{k_1}(V))\otimes \pi_2^*(\gamma^{k_2}(V)^\circ),\mathbb{K}) \cong \pi_1^*(\gamma^{k_1}(V))^*\otimes \pi_2^*(\gamma^{k_2}(V)^\circ)^*,$$
where $\gamma^{k_2}(V)^\circ$ is the "annihilator bundle" of $\gamma^{k_2}(V)$ (a term I could not find in the literature, but the total space conists of pairs $(W,\phi)$ with $W \in G_{k_2}(V)$ and $\phi\in W^\circ$, so I guess this should be a vector bundle of rank $(n-k_2)$ by some standard argument).
The pairing $s(V_1,V_2)$ above should define a section $s$ of this bundle $\mu$. And it vanishes at a point $(V_1,V_2)$ if and only of $V_1\subset V_2$ so it should hold that $$Z(s) = Fl(k_1,k_2).$$
In that case, assuming the section to be transverse to the zero section, we have $$j^*\tau_{G_{k_1}(V)\times G_{k_2}(V)} \cong \tau_{Fl(k_1,k_2)}\oplus j^*\mu,\quad j:Fl(k_1,k_2)\hookrightarrow G_{k_1}(V)\times G_{k_2}(V),$$
so $\mu$ should indeed be a normal bundle.
I guess that I could replace this "annihilator bundle" $\gamma^{k_2}(V)^\circ$ by the dual of the quotient bundle $Q^{k_2}(V)^*$ of $\gamma^{k_2}(V)$ since they should be isomorphic.
My questions:

*

*Is this construction valid or does it have some flaws?

*This construction appears to be very standard knowledge, but I could not find similar arguments somewhere. Does anyone have any references? There should be a standard way to construct a normal bundle of flag varieties...

Thank you very much in advance!
 A: Question: "Is this construction valid or does it have some flaws?
This construction appears to be very standard knowledge, but I could not find similar arguments somewhere. Does anyone have any references? There should be a standard way to construct a normal bundle of flag varieties..."
Answer: You should look up the definition of the normal bundle: If $i:X \rightarrow Y$ is a regular embedding (over $S$), there is a sequence
$$K \rightarrow i^*\Omega^1_{Y/S} \rightarrow \Omega^1_{X/S} \rightarrow \Omega^1_{X/Y} \rightarrow 0$$
And $K$ is a finite rank locally trivial sheaf with $N_X(Y):=\mathbb{V}(K^*)$ the associated vector bundle.
In your case it follows $Y:=G_1\times G_2$ is a product of grassmannians and if
$$0 \rightarrow S_1\rightarrow \pi_1^*V \rightarrow Q_1 \rightarrow 0$$ is the tautological sequence on $G_1$ it follows
$$\tau_{G_1} \cong Hom (S_1,Q_1) \cong S_1^* \otimes Q_1$$
and
$$\Omega^1_{G_1/k} \cong S_1 \otimes Q_1^*.$$
It follows
$$\Omega^1_{G_1 \times G_2/k} \cong p_1^*\Omega^1_{G_1/k}\oplus p_2^*\Omega^1_{G_2/k}.$$
It follows
$$CF.\text{  }i^*\Omega_{Y/k}\cong q_1^*\Omega^1_{G_1/k} \oplus q_2^*\Omega^1_{G_2/k}$$
is an isomorphism where
$$q_i: F \rightarrow G_i$$
is the canonical "projection map" and $F:=F(k_1,k_2)$ is the flag variety. You get a sequence
$$ K \rightarrow q_1^*\Omega^1_{G_1/k}\oplus q_2^*\Omega^1_{G_2/k} \rightarrow \Omega^1_{F/k}$$
and this defines the normal bundle $N_F(G_1\times G_2)$. You must compare this construction with your construction. There is for the flag bundle $F$ a tautological flag
$$T1.\text{   } F_1 \subseteq F_1 \subseteq \pi^*V$$
where $\pi: F \rightarrow Spec(k)$ is the projection map and $\pi^*V$ is the trivial bundle on $F$ of rank $dim_k(V)$. If $x:=[V_1 \subseteq V_2]\in F(k)$ is a $k$-rational point, it follows
$$F_1(x)=V_1 \subseteq F_2(x)=V_2 \subseteq V$$
when you pass to the fiber of $T1$ at $x$. This is similar to what happens to the grassmannian $G_1$: If $x:=[V_1]\in G_1(k)$ is a point corresponding to $W \subseteq V$ you get the exact sequence
$$0 \rightarrow S_1(x)\cong V_1 \rightarrow  V \rightarrow V/V_1 \rightarrow 0$$
gives back the $k_1$-dimensional $k$-vector space $V_1 \subseteq V$.
The fiber of $\Omega^1_{G_1/k} \cong S_1\otimes Q_1^*$ at $x$ is the vector space
$$S_1\otimes Q_1^*(x) \cong V_1 \otimes (V/V_1)^*.$$
Hence the fiber of $i^*\Omega^1_{G_1 \times G_2/k}$ at the $k$-rational point $y:=[V_1 \subseteq V_2]$ is the vector space
$$ V_1\otimes(V/V_1)^* \oplus V_2\otimes(V/V_2)^*.$$
You must interpret this in terms of the map $i^*\Omega^1_{G_1 \times G_2/k} \rightarrow \Omega^1_{F/k}$ and identify the kernel $K$.
Comment: "I guess that I could replace this "annihilator bundle" γk2(V)∘ by the quotient bundle Qk2(V) of γk2(V) since they should be isomorphic."
Response: I have not seen the "annihilator bundle" mentioned. There is a canonical map
$$<,>: V_2 \times V^* \rightarrow k$$
defined by
$$<v,\phi>:=\phi(v)$$
inducing a map
$$ f: V_2 \otimes_k V^* \rightarrow k$$
by $f(v\otimes \phi):=\phi(v)$. This induce a map
$$f^*: V^* \rightarrow Hom_k(V_2,k)$$
and $V_2^{\bullet}:=ker(f^*)$. You should formulate this notion precisely in terms of bundles and the tautological sequences on the grassmannian variety and flag variety. In fact for the grassmannian $G(k_2,V)$ there is the "dual" tautological sequence
$$0 \rightarrow Q_2^* \rightarrow \pi_2^*V^* \rightarrow S_2^* \rightarrow 0$$
and it may be $(S_2)^{\bullet}\cong Q_2^*$ is the dual of the quotient bundle $Q_2$. You should try to use the bundle $Q_2^*$ instead of the "annihilator bundle". From my argument you see that $Q_i^*$ is involved in formula $CF$ when calculating the cotangent bundle of the grassmannian and the flag variety.
My calculation gives
$$j^*\tau_{G_1 \times G_2/k}=f_1^*(S_1^*\otimes Q_1) \oplus f_2^*(S_2^*\otimes Q_2)$$
and
$$ j^*\mu=q_1^*(S_1^*)\otimes q_2^*(Q_2)$$
where $q_i: F(k_1,k_2) \rightarrow G_i$ is the canonical map $q_i([V_1 \subseteq V_2]):=[V_i] \in G_i$.
Note: There is another map that may be interesting:
$$f: F(k_1,k_2) \rightarrow G(k_1,V)\times G(k_2-k_1,V/V_1)$$
defined by
$$f([V_1,V_2]:=([V_1],[V_2/V_1]).$$
If you construct $F:=F(k_1,k_2):=SL(V)/P$ where $P$ is the parabolic subgroup fixing the flag, it follows
$$dim_k(\Omega^1_{F/k}(e)) =dim_k(Lie(SL(V))/Lie(P))=k_1(n-k_1)+(k_2-k_1)(n-k_2)$$
In algebraic geometry a reference is
Grothendieck, Alexander; Dieudonné, Jean A.
Éléments de géométrie algébrique. I. (English) Zbl 0203.23301
Die Grundlehren der mathematischen Wissenschaften. 166. Berlin-Heidelberg-New York: Springer-Verlag. IX, 466 p. (1971).
There is a book by Akhiezer
Akhiezer, Dmitri N.
Lie group actions in complex analysis. (English) Zbl 0845.22001
Aspects of Mathematics. E27. Braunschweig: Vieweg. vii, 201 p. (1995).
where flag varieties are studied from the point of view of complex analysis.
The quotient-approach: There is another construction of the flag variety and vector bundles using quotients and representations. If $0\neq V_1 \subsetneq \cdots \subsetneq V_d \subsetneq V$ is a flag in $V$, let $P \subseteq SL(V)$ be the parabolic subgroup fixing the flag and construct the "quotient" $SL(V)/P$. (In algebraic geometry this quotient construction is "complicated".) Given any left $P$-module $\rho:P\times W \rightarrow W$ you may construct the associated vector bundle $\pi:V(\rho) \rightarrow SL(V)/P$. If you do this with the $P$-module $T:=Lie(SL(V))/Lie(P)$ you arrive at the tangent bundle $T_{SL(V)/P)}$. If you do this with the dual $T^*$ you arrive at the cotangent bundle $\Omega^1_{SL(V)/P}$. I believe there are references in Akhiezer's book.
