Around the Mukai conjecture for Fano manifolds

TL;DR

This paper explores a generalized Mukai conjecture for Fano manifolds, proposing that those satisfying a specific numerical inequality have unique structures, and classifies such manifolds under certain conditions.

Contribution

It introduces a new conjecture extending the Mukai conjecture and classifies Fano manifolds with Picard number up to 3 or dimension up to 5 that meet the conjecture's criteria.

Findings

Classified Fano manifolds with  ho_X 3 or 5 dimensions.

Identified special structures for manifolds satisfying the conjecture.

Extended understanding of the structure of Fano manifolds under the generalized Mukai conjecture.

Abstract

As a generalization of the Mukai conjecture, we conjecture that the Fano manifolds $X$ which satisfy the property $\rho_X(r_X-1)\geq\dim X-1$ have very special structure, where $\rho_X$ is the Picard number of $X$ and $r_X$ is the index of $X$. In this paper, we classify those $X$ if $\rho_X\leq 3$ or $\dim X\leq 5$.