A remark on Ulrich and ACM bundles

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 BX1 of locally exact differentials twisted by OX(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 BX1 is an ACM bundle and if X is also a Calabi–Yau variety and p>2 then BX1 is not a direct sum of line bundles. In particular I show that BX1 is an ACM bundle on any ordinary Calabi–Yau variety. I also prove a characterization of projective varieties with trivial canonical bundle such that BX1 is ACM (for some projective embedding datum): all such varieties are Frobenius split (with trivial canonical bundle).