Foliations and Purely Inseparable Coverings

Overview

In Lecture II, we gave a numerical characterization of uniruled varieties in terms of a certain numerical property of anti-canonical divisors. In this lecture, we discuss a refined characterization of such varieties in terms of the tangent bundle. Namely, a smooth projective variety in characteristic zero is uniruled unless its tangent bundle is almost everywhere seminegative.

The proof of this result is made by using quotient varieties by foliations in positive characteristics. Since tangent bundles in positive characteristics, viewed as sheaves of derivations, have features quite different from those in characteristic zero, we discuss some properties of derivations and differential operators in characteristic p in detail. Then a variety in characteristic p is shown to be uniruled if it carries a foliation (a subsheaf which is closed under Lie bracket and p-th power) with certain properties.

What relates this new criterion of uniruledness in characteristic p to the one in characteristic zero is “semistability”. The theory of semistable torsion free sheaves will be discussed in the second section, including numerical characterizations of semistability (in characteristic zero) and its behaviour under modulo p reductions. A renowned theorem of Mehta-Ramanathan plays an essential rôle in our argument.

There are several intriguing implications of the refined characterization of uniruledness above. One of them is the result that the 2-cocycle C₂(X) of a minimal variety X looks like an effective cycle, which is derived from the famous Bogomolov inequality for semistable bundles.