Birational boundedness of rationally connected Calabi-Yau 3-folds

TL;DR

This paper establishes the birational boundedness of rationally connected Calabi-Yau 3-folds with certain singularities, advancing understanding of their classification and boundedness properties in algebraic geometry.

Contribution

It proves birational boundedness for rationally connected Calabi-Yau 3-folds with klt singularities and extends results to $ ext{epsilon}$-CY types and log Calabi-Yau pairs.

Findings

Rationally connected Calabi-Yau 3-folds with klt singularities are birationally bounded.

Sets of $ ext{epsilon}$-lc log Calabi-Yau pairs with bounded coefficients are log bounded modulo flops.

Rationally connected klt Calabi-Yau 3-folds with mld bounded away from 1 are bounded modulo flops.

Abstract

We prove that rationally connected Calabi--Yau 3-folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected $3$-folds of $\epsilon$-CY type form a birationally bounded family for $\epsilon>0$. Moreover, we show that the set of $\epsilon$-lc log Calabi--Yau pairs $(X, B)$ with coefficients of $B$ bounded away from zero is log bounded modulo flops. As a consequence, we deduce that rationally connected klt Calabi--Yau $3$-folds with mld bounded away from $1$ are bounded modulo flops.