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We classify the Ulrich vector bundles of arbitrary rank on smooth projective varieties of minimal degree. In the process, we prove the stability of the sheaves of relative differentials on rational scrolls.
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We assume that $mathcal{E}$ is a rank $r$ Ulrich bundle for $(P^n, mathcal{O}(d))$. The main result of this paper is that $mathcal{E}(i)otimes Omega^{j}(j)$ has natural cohomology for any integers $i in mathbb{Z}$ and $0 leq j leq n$, and every Ulrich bundle $mathcal{E}$ has a resolution in terms of $n$ of the trivial bundle over $P^n$. As a corollary, we can give a necessary and sufficient condition for Ulrich bundles if $n leq 3$, which can be used to find some new examples, i.e., rank $2$ bundles for $(P^3, mathcal{O}(2))$ and rank $3$ bundles for $(P^2, mathcal{O}(3))$.
We prove that the kernel bundle of the evaluation morphism of global sections, namely the syzygy bundle, of a sufficiently ample line bundle on a smooth projective variety is slope stable with respect to any polarization. This settles a conjecture of Ein-Lazarsfeld-Mustopa.
We show various properties of smooth projective D-affine varieties. In particular, any smooth projective D-affine variety is algebraically simply connected and its image under a fibration is D-affine. In characteristic zero such D-affine varieties are also uniruled. We also show that (apart from a few small characteristics) a smooth projective surface is D-affine if and only if it is isomorphic to either ${mathbb P}^2$ or ${mathbb P}^1times {mathbb P}^1$. In positive characteristic, a basic tool in the proof is a new generalization of Miyaokas generic semipositivity theorem.
We show that any polarized K3 surface supports special Ulrich bundles of rank 2.
We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6.
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