Existence of Ulrich bundles on all smooth projective varieties

Determine whether every smooth projective variety over an algebraically closed field admits an Ulrich bundle.

Background

Ulrich bundles are vector bundles with strong cohomological vanishing properties that are central to questions about syzygies and projective embeddings. They can be viewed as maximally generated Cohen–Macaulay modules and have been extensively studied on specific classes of varieties, including hypersurfaces via Clifford algebra techniques.

Despite significant progress in special cases and constructions, the general existence problem—asking whether every smooth projective variety supports at least one Ulrich bundle—remains unresolved and is a standard open question in the field. The paper situates its relative results within this broader context.