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

TL;DR

This paper determines the rational homotopy type of the projectivization of the complex tangent bundle over complex projective spaces, revealing its equivalence to a specific homogeneous space.

Contribution

It explicitly computes the rational homotopy type of the total space of the projectivized tangent bundle over complex projective space, a previously uncharacterized space.

Findings

The total space has the rational homotopy type of U(n+1)/U(1)×U(1)×U(n-1).

Provides a clear description of the rational homotopy type of the projectivized tangent bundle.

Advances understanding of the topology of complex vector bundle projectivizations.

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 $\mathrm{U}(n+1)/\mathrm{U}(1)\times \mathrm{U}(1)\times \mathrm{U}(n-1)$.