Why is the Complete Flag Variety an algebraic variety?

Question

Let $V$ be a $\mathbb C $ - vector space of dimension $n$. Let's consider the set $Fl(n)$ of all the complete flags $F_{\bullet}$ $$F_1 \subset F_2 \cdots \subset F_n$$ where the $F_i$ are subspaces of $V$ with $\dim(F_i)=i$ for every $1 \leq i \leq n$.

Why is this an affine/projective variety? I know that given that we can use the transitive action of $GL(n, \mathbb C)$ and deduce that $$Fl(n) \simeq GL(n, \mathbb C)/B_n$$ where $B_n$ is the subgroup of the upper triangular matrices. But first we need to show that it is an algebraic variety.

Thanks!

Answer 1

because of that identification we can put a variety (projective) structure on it...comes from projective structure of $\textrm{GL}(n,\mathbb{C})/B_n$.

there is another identification that this collection of complete flags can be thought of as inside product of $\mathbb{G}(1,n) \times \mathbb{G}(2,n) \times \cdots \times \mathbb{G}(n-1,n)$ where $\mathbb{G}(r,n)$ Grassmanian variety .

Answer 2

Question: "Why is this an affine/projective variety?"

Answer: If $X \subseteq \mathbb{P}^n_k$ is a projective scheme over a field $k$ and if $E$ is a finite rank locally trivial sheaf on $X$, it follows $\mathbb{P}(E^*)$ is again a projective scheme of finite type over $k$. The complete flag scheme/variety $F(\underline{d},E)$ of $E$ may be realized as a sequence of such projective bundles: There is a finite sequence

$$F(\underline{d},E) \cong Y_n \rightarrow Y_{n-1} \rightarrow \cdots \rightarrow Y_1 \rightarrow^{\pi_1} X$$

where each morphism $\pi_{i}: Y_i \rightarrow Y_{i-1}$ is a projective bundle. Hence by induction: $F(\underline{d},e)$ is a projective scheme of finite type over $k$.

Example: If $V:=k\{e_0,..,e_n\}$ is a vector space over $k$ and $X:=Spec(k)$, it follows $F(\underline{d},V)$ is the complete flag variety of the vector space $V$. If $G:=SL(V)$ and $P \subseteq G$ is a parabolic subgroup of type $\underline{d}$ it follows $G/P \cong F(\underline{d},V)$.

Example: Using the projective bundle formula and induction you may calculate the Chow group and Grothendieck group (and many other cohomology groups) of $F(\underline{d},E)$:

$$ K_0(\mathbb{P}(E^*)) \cong K_0(X)[t]/(t^{e+1})$$

where $rk(E)=e+1$.