A remark on Ulrich and ACM bundles

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).