The generalised Mukai conjecture for spherical varieties

Abstract

We prove the generalised Mukai conjecture for $\mathbb{Q}$-factorial spherical Fano varieties. In this case, a stronger inequality holds featuring an extra term - the minimum absolute complexity of a log Calabi-Yau pair - which measures how close the Fano variety is to being toric.

A question of Gongyo on nef partitions is answered. There are new examples, and the previous version has been split into two separate papers