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Consider the Euler exact sequence:

$ 0\longrightarrow \mathcal{O}_{\mathbb{P}^n} \longrightarrow \mathcal{O}_{\mathbb{P}^n}(1)^{n+1}\longrightarrow \mathcal{T}_{\mathbb{P}^n} \longrightarrow 0 $

This tells me that for example, the vector fields in the projective space are of the form $ v^i\partial_i $ with $v^i$ linear in the coordinates $x_i$ of $\mathbb{C}^{n+1}$. I want to study the space $\bigwedge^2\mathcal{T}_{\mathbb{P}^n}$. How can I construct a second exterior power of the Euler sequence to study that?

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    $\begingroup$ See for instance Hartshorne, Exercise II.5.16 for turning short exact sequences of vector bundles into filtrations computing their symmetric or exterior powers. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 10, 2023 at 0:51
  • $\begingroup$ thanks, if I well understood, we have $ \bigwedge^2\mathcal{T}_{\mathbb{P}^n} \cong \frac{\bigwedge^2\mathcal{O}(1)^{n+1}}{\mathcal {T}_{\mathbb {P}^n} } $ $\endgroup$
    – BVquantization
    Commented Feb 10, 2023 at 2:26
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    $\begingroup$ Ah right, in this case $\bigwedge^i \mathcal O_{\mathbf P^n} = 0$ for $i \geq 2$, so you just get a bunch of short exact sequences $0 \to \bigwedge^{i-1}\mathcal T_{\mathbf P^n} \to \bigwedge^i \mathcal O_{\mathbf P^n}(1)^{n+1} \to \bigwedge^i \mathcal T_{\mathbf P^n} \to 0$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 10, 2023 at 3:49

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