Obstructions to the ampleness of invariant jet differentials on complete intersections

Abstract

We show that on every smooth hypersurface $X\subset\mathbb P^4$ the Demailly--Semple bundle $E_{3,5q}T^*_X$ of invariant jet differentials of order $3$ and weighted degree $5q$ is not ample, for every integer $q\ge1$. This contradicts the Diverio--Trapani conjecture in the form ``$E_{k,m}T^*_X$ is ample for all $m\gg0$'' 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 $E_{k,m}T^*_X$ depends on $m$.