Global differential forms on Fano varieties

Question

Let $X$ be a smooth rationally connected variety (say in characteristic zero). Then it is straightforward to prove that $H^0(X,(\Omega_X^1)^{\otimes m}) = 0$ for all $m > 0$ since there is a very free rational curve $f: \mathbb{P}^1 \rightarrow X$ for which $f^* T_X$ is ample (see for example Rational Curves on Algebraic Varieties by J. Kollár, chapter IV). In fact, the vanishing of these cohomology groups conjecturally characterizes rationally connected varieties.

However, this doesn't seem to immediately rule out the existence of a nonzero global $m$-form for some $m > 1$, since $\Omega^m_X$ is a quotient of $(\Omega_X^1)^{\otimes m}$.

Question. Are there any rationally connected (or even Fano) varieties with nonzero global differential $m$-forms?

For $X$ smooth Fano of dimension $n$, we would necessarily have $1 < m < n$.

Answer 1

For Fano manifolds in characteristic $0$, we can also use the following argument.

By Dolbeault theorem and Hodge symmetry we get, for any compact Kähler manifold $X$ and for any $m>0$, the isomorphism $$H^0(X, \, \Omega^m_X)\simeq H^m(X, \, \mathcal{O}_X) =H^m(X, \, K_X \otimes K_X^{-1}).$$ If $X$ is Fano then $K_X^{-1}$ is ample and so the last group is zero by Kodaira vanishing theorem.