The dual complex of log Calabi-Yau pairs on Mori fibre spaces

Abstract

In this paper we show that the dual complex of a dlt log Calabi-Yau pair $(Y, \Delta)$ on a Mori fibre space $\pi: Y \to Z$ is a finite quotient of a sphere, provided that either the Picard number of $Y$ or the dimension of $Z$ is $\leq 2$. This is a partial answer to Question 4 in "The dual complex of Calabi-Yau pairs" by Koll\'ar and Xu.