Some evidence for the existence of Ulrich bundles
Stefan Deaconu
TL;DR
The paper investigates the existence of Ulrich bundles on nonsingular projective varieties by presenting two complementary frameworks. A K-theoretic route uses $K_{0,\mathbb{Q}}$-classes, motivic vector bundles, and Jouanolou devices to lift classes along finite surjections and derive Ulrich bundles on the original variety under favorable conditions (notably for $\dim X\le 3$). A derived-category route introduces derived Ulrich objects relative to ample sequences and shows how top nonzero cohomology yields Ulrich objects or bundles, with preservation under hyperplane sections and finite morphisms. The work yields corollaries such as a K-theoretic Ulrich bundle if motivic bundles are algebraic and a procedure to obtain Ulrich bundles on surfaces from derived Ulrich objects, offering constructive pathways toward linearizing Chow forms and enriching moduli perspectives. Together, these approaches provide evidence for the existence of Ulrich bundles and establish frameworks connecting K-theory, motivic theory, and derived categories to extrinsic questions in algebraic geometry.
Abstract
The question of existence of Ulrich bundles on nonsingular projective varieties is posed here in weaker terms: either to find a K-theoretic solution, or to find one in the derived category of the variety. We observe that if any motivic vector bundle is algebraic, there is always a solution in the Grothendieck group. Also, by considering the derived problem, it is noted a formal way of producing Ulrich sheaves on a surface.