Abstract
We prove that smooth, projective, K-trivial, weakly ordinary varieties over a perfect field of characteristic p>0 are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our work, together with Langer’s results, implies that varieties of the above type have strongly semistable tangent bundles with respect to every polarization.
Highlights
We work over a perfect field k of characteristic p > 0
Consider a smooth projective K -trivial variety X of dimension d defined over k, where K -trivial means that K X ∼ 0
It is inherent for our methods that we prove a singular version of Theorem 1.1 too: Theorem 1.2 (Theorem 5.6) Singular case: Let X be a normal, S3, projective, globally F-split variety over k with W O-rational singularities and with K X ∼ 0
Summary
We work over a perfect field k of characteristic p > 0. If X is rational, that is, there exists a resolution of singularities f : Y → X such that R f∗OY = OX , or if X is a klt threefold (see [18, Theorem 1.4]) We finish this part of the introduction by noting that by the previous work of the authors [39], a Beauville–Bogomolov type decomposition holds for K -trivial weakly ordinary varieties. We remark that in characteristic zero the full decomposition theorem relies on stability properties of the tangent bundle In this spirit, using recent result of Langer [29, Corollary 3.3] we obtain the following: Corollary 1.5 If X is a smooth projective weakly ordinary K X ≡ 0), the tangent sheaf TX is strongly H -semistable for every ample divisor H on X
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
