The differential-form criterion for rational connectedness

Let $X$ be a smooth projective variety. Write $\Omega^1_X$ for the sheaf of differential $1$-forms on $X$, and $(\Omega^1_X)^{\otimes m}$ for its $m$-fold tensor power.

Rational connectedness criterion. $X$ is rationally connected if and only if

$$H^0(X,(\Omega^1_X)^{\otimes m})=0$$

for all $m>0$.

This is proposed as a numerical criterion for rational connectedness, expressing the absence of global tensorial differential forms in terms of connectivity by rational curves. Its resolution is not specified in the supplied text.