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 $H^0(X,\Omega\_X^q(q))=0$ ?



Sections of a sheaf of differentials on a weighted complete intersection
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.


