Line-filtered ACM bundles and Ulrich defects on abelian varieties of Picard rank one
Abstract
Let A be a complex abelian variety with NS(A)=Z[L], L ample. We show that a vector bundle filtered by line bundles splits uniquely into unipotent blocks, determine when it is ACM with respect to L, and show that it is never Ulrich. For a rank-two Ulrich bundle we bound from below the degree of the defect of every saturated line subbundle, compute the Hilbert polynomial of the defect, and show that the bound is attained exactly when an explicit cohomological support locus is nonempty. On abelian surfaces this locus is an ample curve in Pic0(A); as a consequence, every rank-two Ulrich bundle is a Serre extension of a fixed numerical type, which we characterize and construct for every ample L. For g≥3, we reduce the equality case to the existence of subcanonical codimension-two subschemes with prescribed cohomology.
Related
- On the existence of smooth varieties of any dimension carrying no Ulrich bundles for some very ample line bundle
- Existence of Ulrich bundles on polarized surfaces and weakly Ulrich bundles on polarized varieties in positive characteristic
- On the existence of polarized varieties without weakly Ulrich bundles
- Ampleness of non-symmetric line bundles on degenerations of abelian varieties
- Ulrich Bundles on certain Grassmann Bundles Over Curves
BibTeX
@misc{mondal2026linefiltered,
title = {{Line-filtered ACM bundles and Ulrich defects on abelian varieties of Picard rank one}},
author = {Soham Mondal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.09163},
eprint = {2608.09163},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09163},
}
RIS
TY - UNPB
TI - Line-filtered ACM bundles and Ulrich defects on abelian varieties of Picard rank one
AU - Soham Mondal
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2608.09163
UR - https://arxiv.org/abs/2608.09163
AB - Let A be a complex abelian variety with NS(A)=Z[L], L ample. We show that a vector bundle filtered by line bundles splits uniquely into unipotent blocks, determine when it is ACM with respect to L, and show that it is never Ulrich. For a rank-two Ulrich bundle we bound from below the degree of the defect of every saturated line subbundle, compute the Hilbert polynomial of the defect, and show that the bound is attained exactly when an explicit cohomological support locus is nonempty. On abelian surfaces this locus is an ample curve in Pic^0(A); as a consequence, every rank-two Ulrich bundle is a Serre extension of a fixed numerical type, which we characterize and construct for every ample L. For g≥3, we reduce the equality case to the existence of subcanonical codimension-two subschemes with prescribed cohomology.
ER -