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Given a smooth hypersurface $X \subset \mathbb{P}^{2p+1}_{\mathbb{C}}$ of degree $d \ge 5$ (with $p\ge 2$), when can we say that

$H^{p+1}(\mathcal{O}_X)=H^{p+1}(\Omega^1_X)=...=H^{p+1}(\Omega^{p-2}_X)=0$ ?

Are there some examples of such a hypersurface? Any suggestions on a possible direction of approaching this problem is more than welcome.

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 Maybe I misunderstand something basic but I think the weak Lefschetz theorem implies that these groups are always $0$ (for any $d>0$). – Damian Rössler Aug 19 at 12:44
That can't be right: if $p=1$ and $d=4$, we are talking about a smooth quartic $X$ in $\mathbb{P}^3$, for which $h^{p+1}(O_X)=h^{0,2}(X)=1$. – René Pannekoek Aug 19 at 12:54
@René Pannekoeck: sorry: to apply the Lefschetz hyperplane theorem, the condition $p+1+i<2p$ must be satisfied for $i=0,\dots,p-2$. So if $p<2$, you have a problem. So what I am saying is true if $p>1$. – Damian Rössler Aug 19 at 13:36
(note that the statement of the question is ambiguous if $p<2$) – Damian Rössler Aug 19 at 13:39
@Rossler: Thanks a lot for the answer. I am sorry I forgot to specify but $p \ge 2$ in the question. – Naga Venkata Aug 19 at 14:30

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