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<p>We determine the rational homotopy type of the total space of the projectivization of the complex tangent bundle <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\tau : \mathbb {C}^n \longrightarrow E \longrightarrow \mathbb {C}P^n\)</EquationSource> </InlineEquation>. We show that the total space <i>P</i>(<i>E</i>) of the projectivization bundle <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(P(\tau ): \mathbb {C}P^{n-1} \longrightarrow P(E) \longrightarrow \mathbb {C}P^n\)</EquationSource> </InlineEquation> has the rational homotopy type of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(U(n+1)/U(1) \times U(1) \times U(n-1)\)</EquationSource> </InlineEquation>, where <i>U</i>(<i>k</i>) is the unitary group.</p>

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The rational homotopy type of the projectivization of the tangent bundle of complex projective spaces

  • 期刊论文

  • 发表日期：

    2025年11月20日

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  • Meshach Ndlovu,
  • Jean Baptiste Gatsinzi

摘要

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 \(U(n+1)/U(1) \times U(1) \times U(n-1)\) , where U(k) is the unitary group.