Obstructions to the ampleness of invariant jet differentials on complete intersections

We show that on every smooth hypersurface $X\subset {\mathbb{P}}^4$ the Demailly–Semple bundle $E_{3,5q}{T}_X^{\ast }$ of invariant jet differentials of order $3$ and weighted degree $5q$ is not ample, for every integer $q\ge 1$. This contradicts the Diverio–Trapani conjecture in the form ``$E_{k,m}{T}_X^{\ast }$ is ample for all $m\gg 0$’’ for hypersurfaces in ${\mathbb{P}}^4$ at $k=3$. We then extend the obstruction to complete intersections of any dimension, to jet differentials of order $4$, and to all sufficiently divisible weights, and we discuss how ampleness of $E_{k,m}{T}_X^{\ast }$ depends on $m$.