The generalised Mukai conjecture for spherical varieties

TL;DR

This paper proves the generalized Mukai conjecture for a class of spherical Fano varieties, introducing a stronger inequality that incorporates a measure of how close these varieties are to being toric.

Contribution

The paper establishes the conjecture for $Q$-factorial spherical Fano varieties and introduces a new inequality involving the minimum absolute complexity of a log Calabi-Yau pair.

Findings

Proof of the generalized Mukai conjecture for $Q$-factorial spherical Fano varieties

Introduction of a stronger inequality involving absolute complexity

Quantitative measure of how close a Fano variety is to being toric

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.