Almost nef regular foliations and Fujita's decomposition of reflexive sheaves

TL;DR

This paper characterizes almost nef regular foliations on smooth projective varieties, generalizes Fujita's decomposition theorem, and explores the structure of reflexive sheaves and foliations with nef anti-canonical bundles.

Contribution

It provides a structure theorem for almost nef regular foliations, generalizes Fujita's decomposition, and links these to the geometry of reflexive sheaves and algebraic fiber spaces.

Findings

Almost nef tangent bundle of a rationally connected variety is generically ample.

Reflexive hull of pushforward of multiple canonical divisors decomposes into flat and ample parts.

Foliations with nef anti-canonical bundles are studied in depth.

Abstract

In this paper, we study almost nef regular foliations. We give a structure theorem of a smooth projective variety $X$ with an almost nef regular foliation $\mathcal{F}$: $X$ admits a smooth morphism $f: X \rightarrow Y$ with rationally connected fibers such that $\mathcal{F}$ is a pullback of a numerically flat regular foliation on $Y$. Moreover, $f$ is characterized as a relative MRC fibration of an algebraic part of $\mathcal{F}$. As a corollary, an almost nef tangent bundle of a rationally connected variety is generically ample. For the proof, we generalize Fujita's decomposition theorem. As a by-product, we show that a reflexive hull of $f_{*}(mK_{X/Y})$ is a direct sum of a hermitian flat vector bundle and a generically ample reflexive sheaf for any algebraic fiber space $f : X \rightarrow Y$. We also study foliations with nef anti-canonical bundles.