How can one prove that the tangent bundle $T_{\mathbb{P}^2}$ and its dual $\Omega_{\mathbb{P}^2}$ are stable vector bundles with respect to $\mathcal{O}_{\mathbb{P}^2}(1)$? Similarly, is it true that $T_{\mathbb{P}^n}$ and its dual $\Omega_{\mathbb{P}^n}$ are stable vector bundle for any $n \in \mathbb{N}$? By stable I mean polynomial stability.
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
For $P^2$ it is enough to check that $H^0(T(-2)) = 0$ which follows immediately from the Euler sequence. For higher $n$ one should check that $H^0(\Lambda^kT(-k-1)) = 0$ which follows from (the exterior power of) the Euler sequence.
-
$\begingroup$ Thank you for the answer, but could you explain a bit more? I don't know why it is enough to check $H^0(T_{\mathbb{P}^2}(-2))=0$. I am not familiar with this field. $\endgroup$– user2013Mar 21, 2013 at 23:20