Existence of complements for foliations

TL;DR

This paper proves the existence of rational complements for certain algebraically integrable foliations on specific varieties and explores related properties like inversion of adjunction and connectedness.

Contribution

It establishes the existence of $Q$-complements for algebraically integrable log-Fano foliations and studies their key properties.

Findings

Existence of $Q$-complements for algebraically integrable log-Fano foliations.

Partial inversion of adjunction for these foliations.

Connectedness principle related to the foliations.

Abstract

This paper demonstrates the existence of $\mathbb{Q}$-complements for algebraically integrable log-Fano foliations on klt ambient varieties. Additionally, we investigate properties of algebraically integrable Fano foliations such as a partial inversion of adjunction as well as a connectedness principle.