Boundedness of complements for log Calabi-Yau threefolds

TL;DR

This paper proves the existence of a finite set of integers bounding complements for Calabi-Yau threefold pairs with certain singularities, extending the theory of complements in algebraic geometry.

Contribution

It establishes the boundedness of complements for log Calabi-Yau threefolds and surfaces, providing a finite set of integers for complements in these cases.

Findings

Finite set of positive integers for complements exists.

Boundedness of complements for surface pairs proven.

Extension of complement theory to Calabi-Yau threefolds.

Abstract

In this paper, we study the theory of complements, introduced by Shokurov, for Calabi-Yau type varieties with the coefficient set $[0,1]$. We show that there exists a finite set of positive integers $\mathcal{N}$, such that if a threefold pair $(X/Z\ni z,B)$ has an $\mathbb{R}$-complement which is klt over a neighborhood of $z$, then it has an $n$-complement for some $n\in\mathcal{N}$. We also show the boundedness of complements for $\mathbb{R}$-complementary surface pairs.