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Generic vanishing subschemes of codimension three
Abstract
We classify geometrically nondegenerate GV subschemes of codimension three in indecomposable principally polarized complex abelian varieties of dimension g≥6. Every such subscheme is a translate of ± Wg−3(C) in the Jacobian of a smooth curve C. The proof studies the theta-dual surface: tangent cones and theta Hessians give a curve summand in the singular case, while Hodge gradings and characteristic cycles give one in the smooth case. Theta duality yields the corresponding classification of GV surfaces. We also compute the cohomology of all topologically trivial twists of the ideal sheaf twisted by the principal polarization, and characterize the hyperelliptic case by a cohomology jump.
Related
- On finite covers of projective spaces, generic outside of codimension three
- Generic vanishing conjecture for divisors
- Generic vanishing for Dolbeault cohomology of mixed Hodge modules on abelian varieties
- Codimension-three regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound
- Numerical Kawamata--Viehweg vanishing for rank-three matroids
BibTeX
@misc{chen2026generic,
title = {{Generic vanishing subschemes of codimension three}},
author = {Yuesen Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15391},
eprint = {2609.15391},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15391},
}
RIS
TY - UNPB
TI - Generic vanishing subschemes of codimension three
AU - Yuesen Chen
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.15391
UR - https://arxiv.org/abs/2609.15391
AB - We classify geometrically nondegenerate GV subschemes of codimension three in indecomposable principally polarized complex abelian varieties of dimension g≥6. Every such subscheme is a translate of ± W_g−3(C) in the Jacobian of a smooth curve C. The proof studies the theta-dual surface: tangent cones and theta Hessians give a curve summand in the singular case, while Hodge gradings and characteristic cycles give one in the smooth case. Theta duality yields the corresponding classification of GV surfaces. We also compute the cohomology of all topologically trivial twists of the ideal sheaf twisted by the principal polarization, and characterize the hyperelliptic case by a cohomology jump.
ER -