The Grassmannian is a complex manifold

 Lecture topic

Let $Gr(k,\C^n)$ be the space of $k$-dimensional complex subspaces of $\C^n$, also known as the complex Grassmannian. We will show that it is a complex manifold of dimension $k(n-k)$. Thanks to Jinhua Xu and professor Mihai Păun for explaining the details.

To begin, take $P\in Gr(k,\C^n)$ and an $n-k$ subspace $Q$ of $\C^n$, such that $P\cap Q = \{0\}$. Then $P\oplus Q = \C^n$, so we have natural projections

A neighborhood of $P$, depending on $Q$ may be described as $U_Q = \{S\in Gr(k,\C^n)\ :\ S\cap Q = \{0\}\}$. We claim that $U_Q \cong \Hom(P,Q)$. The isomorphism is described by
\[
\begin{array}{r c c c l}
\Hom(P,Q) & \to & U_Q & \to & \Hom(P,Q), \\
A & \mapsto & \{v+Av\ :\ v\in P\}, \\
& & S & \mapsto & \left(\pi_Q|_S\right) \circ \left(\pi_P|_S\right)^{-1}.
\end{array}
\]
The map on the right, call it $\varphi_Q$, is also the chart for the manifold structure. The idea of decomposing $\C^n$ into $P$ and $Q$ and constructing a homomorphism from $P$ to $Q$ may be visualized in the following diagram.

Then $\Hom(P,Q) \cong \Hom(\C^k,\C^{n-k})\cong \C^{k(n-k)}$, so $Gr(k,\C^n)$ is locally of complex dimension $k(n-k)$. To show that there is a complex manifold structure, we need to show that the transition functions are holomorphic. Let $P,P'\in Gr(k,\C^n)$ and $Q,Q'\in Gr(n-k,\C^n)$ such that $P\cap Q = P'\cap Q' = \{0\}$. Let $X\in \Hom(P,Q)$ such that $X\in \varphi_Q(U_Q\cap U_{Q'})$, with $\varphi_Q(S)=X$ and $\varphi_{Q'}(S)=X'$ for some $S\in U_Q\cap U_{Q'}$. Define $I_X(v) = v+Xv$, and note the transition map takes $X$ to
\begin{align*}
X' & = \varphi_{Q'}\circ \varphi_Q^{-1}(X) & (\text{definition}) \\
& = \varphi_{Q'}(S) & (\text{assumption})\\
& = \left(\pi_{Q'}|_S\right)\circ \left(\pi_{P'}|_S\right)^{-1} & (\text{definition}) \\
& = \left(\pi_{Q'}|_S\right)\circ I_X\circ I_X^{-1}\circ \left(\pi_{P'}|_S\right)^{-1} & (\text{creative identity}) \\
& = \left(\pi_{Q'}|_S\circ I_X\right)\circ \left(\pi_{P'}|_S\circ I_X\right)^{-1} & (\text{redistribution}) \\
& = \left(\pi_{Q'}|_P +\pi_{Q'}|_Q\circ X\right) \circ \left(\pi_{P'}|_P +\pi_{P'}|_Q\circ X\right). & (\text{definition})
\end{align*}
At this last step we have compositions and sums of homomorphisms and linear maps, which are all holomorphic. Hence the transition functions of $Gr(k,\C^n)$ are holomorphic, so it is a complex manifold.