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I would like to explicitly compute the charts and transition charts for the tangent bundle of $\mathbb{CP}^3$.

I know the charts of $\mathbb{CP}^3$ are

$\phi_i: U_i=\{[z_0,z_1,z_2,z_3]; z_i \neq 0\}\rightarrow \mathbb{R}^2$ where $\phi_i[z_0,z_1,z_2,z_3]=(z_j/z_i)_{j \neq i}$

Is the chart for the tangent bundle

$T\phi_{i}(x,v)=(\phi_i(x),v)$?

Denoting $\phi_{ij}=\phi_j \circ \phi_i^{-1}$ the transition charts, are the transition charts for the tangent bundle

$T\phi_{ij}(z,v)=(\phi_{ij}(z),(D\phi_{ij})_{z}v)$?

Thanks.

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