Boundedness of complements for fibered Fano threefolds in positive characteristic

TL;DR

This paper proves the boundedness of complements for fibered Fano threefolds in large positive characteristic, advancing the understanding of Fano varieties and their fibrations in algebraic geometry.

Contribution

It establishes the canonical bundle formula and confirms Shokurov's conjecture for Fano threefold pairs with specific nef and non-big conditions in large characteristics.

Findings

Proved the canonical bundle formula for Fano type fibrations

Confirmed Shokurov's conjecture in large characteristics under certain conditions

Established boundedness results for complements in positive characteristic

Abstract

In this paper, we prove the canonical bundle formula for Fano type fibrations and Shokurov's conjecture on boundedness of complements for Fano type threefold pairs $(X,B)$ with fibration structures in large characteristics. In particular, we prove the conjecture when $-(K_X+B)\not\equiv 0$ is nef and not big in large characteristics.