Birational complexity of log Calabi-Yau 3-folds

TL;DR

This paper investigates the birational complexity of log Calabi-Yau 3-folds, establishing possible complexity values and describing their birational models with specific contractions, using geometric characterizations.

Contribution

It classifies the birational complexity of certain log Calabi-Yau 3-folds and constructs explicit models with crepant contractions, introducing geometric characterizations of standard -links.

Findings

Birational complexity c_bir(X,B) is in {0, 2, 3}.

Existence of a crepant birational model with a contraction to projective space.

Geometric characterization of standard -links.

Abstract

We study the birational complexity of log Calabi-Yau $3$-folds. For such a pair $(X,B)$ of index one and coregularity zero, we show that $c_{\rm bir}(X,B)\in \{0,2,3\}$. Further, we prove that $(X,B)$ has a log Calabi-Yau crepant birational model that admits a crepant contraction to $(\mathbb{P}^{3-c},H_0+\dots+H_{3-c})$, where $c=c_{\rm bir}(X,B)$. To prove this, we give a geometric characterization of standard $\mathbb{P}^1$-links.