Ordinary varieties with trivial canonical bundle are not uniruled

TL;DR

This paper proves that certain smooth, projective varieties with trivial canonical bundle in characteristic p>0 are not uniruled, and extends the result to some singular cases, impacting the understanding of their tangent bundles.

Contribution

It establishes that smooth, projective, K-trivial, weakly ordinary varieties over fields of characteristic p>0 are not uniruled, including a sharp singular version, and links to semistability of tangent bundles.

Findings

Such varieties are not geometrically uniruled.

A singular version of the theorem is also proven.

Varieties of this type have strongly semistable tangent bundles.

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 any polarization.