Obstructions to the ampleness of invariant jet differentials on complete intersections

Abstract

We introduce the analogue, for invariant jet differentials, of the principal symbol of a differential operator: the component of maximal degree in the highest derivative. The top symbol identifies a canonical quotient bundle of the Demailly--Semple bundle $E_{k,m}{T}_X^{\ast }$, with values in jet differentials of lower order, and together with the Brückmann--Rackwitz vanishing theorem it gives a non-ampleness criterion for $E_{k,m}{T}_Z^{\ast }$ on complete intersections $Z\subset\PP^N$. On the Demailly--Semple tower the top symbol is the restriction to a thickening of the vertical divisor, inside the singular jets. The criterion applies in orders $3$ and $4$: $E_{3,5q}{T}_X^{\ast }$ is not ample on any smooth hypersurface $X\subset\PP^4$, and $E_{4,m}{T}_X^{\ast }$ is not ample for $m\ge 7$ with $m\equiv 0,1,2(\mathrm{mod}7)$ on any smooth hypersurface $X\subset\PP^5$ of degree at least $2$. This contradicts the Diverio--Trapani conjecture at its own threshold.

arXiv commentv2: 36 pages (14 more than v1!). Substantially revised. The Genesis (Section 1), which some readers seem to have enjoyed, now ends on a more optimistic note. New Section 3: the principal symbol of a jet differential and a general non-ampleness criterion (Thm 3.8) of which the main theorems are instances; new Prop 9.2 (E_{3,8} not ample); corrections throughout. Comments very welcome!