Fundamental divisors on Fano varieties of index n-3

TL;DR

This paper generalizes Kawamata's results on the fundamental divisor of Fano manifolds of index n-3, establishing non-vanishing and singularity properties in arbitrary dimensions.

Contribution

It extends Kawamata's findings from four-dimensional cases to all dimensions, proving non-vanishing and singularity bounds for the fundamental divisor.

Findings

Non-vanishing of global sections in arbitrary dimension

General elements have at most canonical singularities

Extension of Kawamata's results to higher dimensions

Abstract

Let X be a Fano manifold of dimension n and index n-3. Kawamata proved the non vanishing of the global sections of the fundamental divisor in the case n=4. Moreover he proved that if Y is a general element of the fundamental system then Y has at most canonical singularities. We prove a generalization of this result in arbitrary dimension.