ACM, almost Ulrich, and Ulrich vector bundles arising from Frobenius
Abstract
Let X be a smooth projective variety of dimension n over an algebraically closed field of positive characteristic. We investigate ACM, almost Ulrich (both ACM and weakly Ulrich), and Ulrich properties of the sheaves of locally exact and closed i-forms BXi, ZXi. These cohomological properties are closely related to the fundamental concepts of Frobenius splitting and ordinarity. Joshi studied the sheaf B1X in this context in \cite{MR3920911, MR4206787}, including ACM and almost Ulrich twists of B1X. However, varieties with Ulrich twisted BX1 have not been classified for n≥2. We classify subcanonical polarized varieties with initialized ACM BiX(aH) for i=1, n and prove that, for n ≥ 2, varieties with ACM BiX for all i ≤ ⌈n/2⌉ are ordinary. When X is ordinary with trivial Ω1X, almost Ulrich twists of BiX and F*e(ZXi(aH)) are determined. We classify polarized varieties (X,H) with BXi(aH) Ulrich when n=1, 2, and establish strong necessary criteria for n ≥ 3, together with an example. We also prove that ZXi(aH) is never Ulrich. Finally, we classify polarized abelian varieties with Ulrich B1X ⊗ L allowing twists by arbitrary line bundles.
Related
- Existence of Ulrich bundles on polarized surfaces and weakly Ulrich bundles on polarized varieties in positive characteristic
- Ulrich Bundles on certain Grassmann Bundles Over Curves
- On the existence of smooth varieties of any dimension carrying no Ulrich bundles for some very ample line bundle
- Ulrich bundles on intersections of two quadrics
- On the classification of polarized surfaces with weakly Ulrich or initialized aCM tangent bundles
BibTeX
@misc{pine2026almost,
title = {{ACM, almost Ulrich, and Ulrich vector bundles arising from Frobenius}},
author = {Jagadish Pine},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28464},
eprint = {2610.28464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28464},
}
RIS
TY - UNPB
TI - ACM, almost Ulrich, and Ulrich vector bundles arising from Frobenius
AU - Jagadish Pine
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28464
UR - https://arxiv.org/abs/2610.28464
AB - Let X be a smooth projective variety of dimension n over an algebraically closed field of positive characteristic. We investigate ACM, almost Ulrich (both ACM and weakly Ulrich), and Ulrich properties of the sheaves of locally exact and closed i-forms B_X^i, Z_X^i. These cohomological properties are closely related to the fundamental concepts of Frobenius splitting and ordinarity. Joshi studied the sheaf B^1_X in this context in \cite{MR3920911, MR4206787}, including ACM and almost Ulrich twists of B^1_X. However, varieties with Ulrich twisted B_X^1 have not been classified for n≥2. We classify subcanonical polarized varieties with initialized ACM B^i_X(aH) for i=1, n and prove that, for n ≥ 2, varieties with ACM B^i_X for all i ≤ ⌈n/2⌉ are ordinary. When X is ordinary with trivial Ω^1_X, almost Ulrich twists of B^i_X and F_*^e(Z_X^i(aH)) are determined. We classify polarized varieties (X,H) with B_X^i(aH) Ulrich when n=1, 2, and establish strong necessary criteria for n ≥ 3, together with an example. We also prove that Z_X^i(aH) is never Ulrich. Finally, we classify polarized abelian varieties with Ulrich B^1_X ⊗ L allowing twists by arbitrary line bundles.
ER -