# Understanding projective spaces as generalized flag varieties

Briefly, is there an intuitive way to understand the relation $$\mathbb{P}^n = \frac{U(n+1)}{U(1)\times U(n)}$$ ? For instance, can one relate it to the usual definition of complex projective spaces as $$\mathbb{P}^n = \left(\mathbb{C}^{n+1} \setminus \{0\}\right) / \mathbb{C}^*$$ ?

• Yes. To identify a space as $G/H$, you find a transitive action of $G$ on it with stabilizer $H$. The action of $U(n+1)$ on $\mathbb{P}^n$ is the obvious one coming from the interpretation of $\mathbb{P}^n$ as classifying lines in $\mathbb{C}^{n+1}$. – Qiaochu Yuan Nov 5 '18 at 23:18
• If you write down the definition of $U(n+1)/(U(1) \times U(n))$ it boils down to look at the first column mod the action of the scalar $U(1)$ i.e it's exactly $\Bbb C^{n+1} \backslash \{0\}/ \Bbb C^*$ – Nicolas Hemelsoet Nov 5 '18 at 23:20