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In this article, I give a crystalline characterization of abelian varieties amongst the class of smooth projective varieties with trivial tangent bundles in characteristic $p>0$. Using my characterization, I show that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion-free. I also show that a conjecture of KeZheng Li about smooth projective varieties with trivial tangent bundles in characteristic $p>0$ is true for smooth projective surfaces. I give a new proof of a result by Li and prove a refinement of it. Based on my characterization of abelian varieties, I propose modifications of Li's conjecture, which I expect to be true. [ABSTRACT FROM AUTHOR]

We examine the relationship between the notion of Frobenius splitting and ordinarity for varieties. We show that the de Rham-Witt cohomology groups Hi(X,W(OX)) of a smooth projective Frobenius split variety are finitely generated over W(k). We provide counterexamples to a conjecture of Mehta that Frobenius split varieties are ordinary or even Hodge-Witt. [ABSTRACT FROM AUTHOR]

A well-known conjecture of Eisenbud, Schreyer and Weyman suggests that any projective variety carries an Ulrich (and hence also weakly Ulrich) bundle. This is known only in a handful of cases. In this paper I provide weakly Ulrich bundles on a class of surfaces and threefolds. I construct a weakly Ulrich bundle of large rank on any smooth complete surface in P 3 over fields of characteristic p > 0 and also for some classes of surfaces of general type in P n. I also construct intrinsic weakly Ulrich bundles on any Frobenius split variety of dimension at most three. The bundles constructed here are in fact ACM and weakly Ulrich bundles and so I call them almost Ulrich bundles. Frobenius split varieties in dimension three include as special cases: (1) smooth hypersurfaces in P 4 of degree at most four, (2) more generally, Frobenius split Fano varieties of dimension at most three, (3) Frobenius split, smooth quintics in P 4 (4) more generally, Frobenius split Calabi-Yau varieties of dimension at most three (5) Frobenius split (i.e. ordinary) abelian varieties of dimension at most three. These results also imply that Chow form of these varieties is the support of a single intrinsic determinantal equation. [ABSTRACT FROM AUTHOR]