Kapranov showed that for a quadric $Q$ in $\mathbb{P}^n$ there is a tilting bundle. For example if we consider a vector space $V$ of dimension $n=4$ and $\mathbb{P}(V)$ then the tilting bundle is $\mathcal{T}=\mathcal{O}\oplus \mathcal{O}(-1)\oplus \sum_{+}(-2)\oplus\sum_{-}(-2)$. Now $\omega^{\vee}_Q=\mathcal{O}(2)$ so what is $\mathrm{Ext}^k(\mathcal{T},\mathcal{T}\otimes \mathcal{O}(2))$? This should be zero for $k>0$...
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$\begingroup$ The case $n = 4$ is quite simple since $Q = P^1\times P^1$ and $\Sigma_\pm$ are $O(1,0)$ and $O(0,1)$ (up to a twist). So, it is very easy to compute the endomorphism algebra. $\endgroup$– SashaCommented Dec 30, 2013 at 12:11
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$\begingroup$ In general you can compute $Ext$'s by Borel--Bott--Weil (all the sheaves are equivariant). $\endgroup$– SashaCommented Dec 30, 2013 at 12:32
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