A remark on Ulrich and ACM bundles

TL;DR

The paper demonstrates the existence of stable Ulrich bundles on ordinary curves and varieties, specifically using the sheaf of locally exact differentials, and characterizes when these bundles are ACM on varieties with trivial canonical bundle.

Contribution

It introduces a natural example of stable Ulrich bundles on ordinary curves and varieties, and characterizes ACM properties of these bundles on Frobenius split Calabi-Yau varieties.

Findings

Existence of stable Ulrich bundles on ordinary curves.

B^1_X is an ACM bundle on Frobenius split varieties.

B^1_X is not a sum of line bundles on Calabi-Yau varieties with p>2.

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 $B^1_X$ 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 $B^1_X$ is an ACM bundle and if $X$ is also a Calabi-Yau variety and $p>2$ then $B^1_X$ is not a direct sum of line bundles. In particular I show that $B^1_X$ is an ACM bundle on any ordinary Calabi-Yau variety. I also prove a characterization of projective varieties with trivial canonical bundle such that $B^1_X$ is ACM (for some projective embedding datum): all such varieties are Frobenius split (with trivial canonical bundle).