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 Ek,mT*X, with values in jet differentials of lower order, and together with the Brückmann--Rackwitz vanishing theorem it gives a non-ampleness criterion for Ek,mT*Z on complete intersections Z⊂PPN. 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 $4$: E3,5qT*X is not ample on any smooth hypersurface X⊂PP4, and E4,mT*X is not ample for m≥7 with m≡0,1,2pmod 7 on any smooth hypersurface X⊂PP5 of degree at least $2$. This contradicts the Diverio--Trapani conjecture at its own threshold.
Related
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BibTeX
@misc{diverio2026obstructions,
title = {{Obstructions to the ampleness of invariant jet differentials on complete intersections}},
author = {Simone Diverio},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33710},
eprint = {2609.33710},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33710},
}
RIS
TY - UNPB
TI - Obstructions to the ampleness of invariant jet differentials on complete intersections
AU - Simone Diverio
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2609.33710
UR - https://arxiv.org/abs/2609.33710
AB - 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,mT^*_X, 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,mT^*_Z on complete intersections Z⊂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_3,5qT^*_X is not ample on any smooth hypersurface X⊂PP^4, and E_4,mT^*_X is not ample for m≥7 with m≡0,1,2pmod 7 on any smooth hypersurface X⊂PP^5 of degree at least $2$. This contradicts the Diverio--Trapani conjecture at its own threshold.
ER -