$\mathcal{O}_X(1)$ be the ample generator of ${\mathop{\rm Pic}\nolimits}X$. Then we have the following vanishing.

1.  $\operatorname{Ext}^q(\mathcal{O}(j), \mathcal{E}(d))=0$ for all $q>0$ and $j< c_1(X)$.

2.  $\operatorname{Ext}^q(\mathcal{O}(c_1(X)), \mathcal{E}(d))=0$ for all $q>0$ if $H(\mathcal{E})^{n+r-1}>0$. If $n=2$ then the condition $H(\mathcal{E})^{n+r-1}>0$ is equivalent to the one that $c_1(\mathcal{E})^2-c_2(\mathcal{E})>0$.

\(1) We have isomorphisms $$\operatorname{Ext}^q(\mathcal{O}(j),\mathcal{E}(d))
\cong H^q(X,\mathcal{E}(d-j))
\cong H^q(\mathbb{P}(\mathcal{E}), H(\mathcal{E})+\pi^*\mathcal{O}_{X}(d-j)).$$ We claim that the last cohomology group vanishes by the Kodaira vanishing theorem; indeed, since $-K_X\cong \mathcal{O}_X(c_1(X))$, we have $$H(\mathcal{E})+\pi^*\mathcal{O}_{X}(d-j)-K_{\mathbb{P}(\mathcal{E})}
\cong (r+1)H(\mathcal{E})+\pi^*\mathcal{O}_{X}(c_1(X)-j),$$ and $(r+1)H(\mathcal{E})+\pi^*\mathcal{O}_{X}(c_1(X)-j)$ is ample by the Nakai-Moishezon criterion, since $j<c_1(X)$. Therefore the claim follows.

\(2) If $j=c_1(X)$, then $H(\mathcal{E})+\pi^*\mathcal{O}_{\mathbb{P}^n}(d-j)-K_{\mathbb{P}(\mathcal{E})}$ is isomorphic to $(r+1)H(\mathcal{E})$, and this is nef and big by assumption. The result then follows from the Kawamata-Viehweg vanishing theorem. The assertion for $n=2$ follows from $H(\mathcal{E})^{r+1}=c_1(\mathcal{E})^2-c_2(\mathcal{E})$ if $n=2$.

\[ProSpaceResol\] Let $\mathcal{E}$ be a nef vector bundle of rank $r$ on $\mathbb{P}^n$. Then $d_{\min}\leqq d$. Moreover if $H(\mathcal{E})^{n+r-1}>0$ then $d_{\min}< d$. In particular if $n=2$ and $c_1(\mathcal{E})^2-c_2(\mathcal{E})>0$, then $d_{\min}< d$.

\[KodairaOnQuadric\] Suppose that $X$ is a smooth hyperquadric $\mathbb{Q}^n$ of dimension $n\geqq 3$. Then $\operatorname{Ext}^q(\mathcal{S},\mathcal{E}(d+j))=0$ for all $q>0$ and $j\geqq -\lfloor\frac{n}{2}\rfloor+1$, where $\mathcal{S}$ is a spinor bundle on $\mathbb{Q}^n$.

We have an isomorphism $\operatorname{Ext}^q(\mathcal{S},\mathcal{E}(d+j))\cong 
H^q(\mathbb{Q}^n, \mathcal{S}^{\vee}\otimes\mathcal{E}(d+j))$. By Theorem \[Usefulottaviani\] (1), (2), and (3), to 