### Abstract

I show that on any smooth, projective ordinary curve of genus at least two and a projective embedding, there is a natural example of a stable Ulrich bundle for this embedding: namely the sheaf B
_{X}
^{1}
of locally exact differentials twisted by O
_{X}
(1) given by this embedding and in particular there exist ordinary varieties of any dimension which carry Ulrich bundles. In higher dimensions, assuming X is Frobenius split variety I show that B
_{X}
^{1}
is an ACM bundle and if X is also a Calabi–Yau variety and p>2 then B
_{X}
^{1}
is not a direct sum of line bundles. In particular I show that B
_{X}
^{1}
is an ACM bundle on any ordinary Calabi–Yau variety. I also prove a characterization of projective varieties with trivial canonical bundle such that B
_{X}
^{1}
is ACM (for some projective embedding datum): all such varieties are Frobenius split (with trivial canonical bundle).

Original language | English (US) |
---|---|

Pages (from-to) | 20-29 |

Number of pages | 10 |

Journal | Journal of Algebra |

Volume | 527 |

DOIs | |

State | Published - Jun 1 2019 |

### Fingerprint

### Keywords

- ACM bundles
- Calabi–Yau variety
- Frobenius split varieties
- Ordinary varieties
- Ulrich bundles

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**A remark on Ulrich and ACM bundles.** / Joshi, Kirti N.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 527, pp. 20-29. https://doi.org/10.1016/j.jalgebra.2019.01.016

}

TY - JOUR

T1 - A remark on Ulrich and ACM bundles

AU - Joshi, Kirti N

PY - 2019/6/1

Y1 - 2019/6/1

N2 - I show that on any smooth, projective ordinary curve of genus at least two and a projective embedding, there is a natural example of a stable Ulrich bundle for this embedding: namely the sheaf B X 1 of locally exact differentials twisted by O X (1) given by this embedding and in particular there exist ordinary varieties of any dimension which carry Ulrich bundles. In higher dimensions, assuming X is Frobenius split variety I show that B X 1 is an ACM bundle and if X is also a Calabi–Yau variety and p>2 then B X 1 is not a direct sum of line bundles. In particular I show that B X 1 is an ACM bundle on any ordinary Calabi–Yau variety. I also prove a characterization of projective varieties with trivial canonical bundle such that B X 1 is ACM (for some projective embedding datum): all such varieties are Frobenius split (with trivial canonical bundle).

AB - I show that on any smooth, projective ordinary curve of genus at least two and a projective embedding, there is a natural example of a stable Ulrich bundle for this embedding: namely the sheaf B X 1 of locally exact differentials twisted by O X (1) given by this embedding and in particular there exist ordinary varieties of any dimension which carry Ulrich bundles. In higher dimensions, assuming X is Frobenius split variety I show that B X 1 is an ACM bundle and if X is also a Calabi–Yau variety and p>2 then B X 1 is not a direct sum of line bundles. In particular I show that B X 1 is an ACM bundle on any ordinary Calabi–Yau variety. I also prove a characterization of projective varieties with trivial canonical bundle such that B X 1 is ACM (for some projective embedding datum): all such varieties are Frobenius split (with trivial canonical bundle).

KW - ACM bundles

KW - Calabi–Yau variety

KW - Frobenius split varieties

KW - Ordinary varieties

KW - Ulrich bundles

UR - http://www.scopus.com/inward/record.url?scp=85062464729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062464729&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2019.01.016

DO - 10.1016/j.jalgebra.2019.01.016

M3 - Article

AN - SCOPUS:85062464729

VL - 527

SP - 20

EP - 29

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -