A note on the rational homotopy type of the projectivization of the tangent bundle of complex projective spaces

Abstract: In this paper, we determine the rational homotopy type of the total space of the projectivization of the complex tangent bundle $\tau : \mathbb{C}^n \longrightarrow E \longrightarrow \mathbb{C}P^{n}$. We show that the total space $P(E)$ of the projectivization bundle $P(\tau): \mathbb{C}P^{n-1} \longrightarrow P(E) \longrightarrow \mathbb{C}P^{n}$ has the rational homotopy type of $\mathrm{U}(n+1)/\mathrm{U}(1)\times \mathrm{U}(1)\times \mathrm{U}(n-1)$.