Let $X\subset\mathbb{P}(a_0,...,a_N)$ be a smooth $n$-dimensional weighted complete intersection in a weighted projective space $\mathbb{P}(a_0,...,a_N)$.
Is it true that if $q\geq 1$ then $H^0(X,\Omega_X^q(q))=0$ ?
2 Answers
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Way no.
Just take a degree $d$ hypersurface in ordinary projective space: $X\subset \mathbb P^n$. Then $$\Omega_X^{\dim X}(\dim X)=\omega_X(n-1)\simeq \mathscr O_X(d-2),$$ so as soon as $d\geq 2$, i.e., it's not a projective space itself, then this sheaf will have non-zero global sections. If you increase the degree you will start to see sections of other $\Omega_X^q(q)$'s. The best you can hope for is a statement such that depending on the degree of the defining equations you might get some vanishing in some range of $q$'s.
answered Nov 5, 2015 at 0:38
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$\begingroup$ Sure, you are right. However in my case $q< \dim(X)$ and I have a problem with complete intersections of quadrics. For instance I know that if $X\subset\mathbb{P}^n$ is a smooth quadric hypersurface then $H^0(X,\Omega_X^q(q))=0$ for $1\leq q\leq \dim(X)-1$. The issue is for a complete intersection of two or more quadrics. $\endgroup$– user75933Commented Nov 7, 2015 at 16:59
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Theorem 0.2 here (On stability of tangent bundles of Fano manifolds with b2 = 1) gives a excellent survey on vanishings. In particular if, as you ask, $q<dim \ X$, then the vanishing properties for $H^0(X, \Omega^q(t))$ holds if $t<q$. So it seems there is no much hope for what you requested.
