Ulrich bundles on cubic fourfolds

Abstract

We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf $\mathcal{E}$ of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an $\mathcal{E}$ appears as an extension of two Lehn–Lehn–Sorger–van Straten sheaves. Then we prove that a general deformation of $\mathcal{E}(1)$ becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6.