Nefness of adjoint bundles for ample vector bundles of corank 3

TL;DR

This paper investigates the conditions under which the adjoint bundle of an ample vector bundle of corank 3 on a smooth projective variety fails to be nef, providing structural insights and implications in adjunction theory.

Contribution

It characterizes pairs (X, E) with ample vector bundle E of rank n-3 where the adjoint bundle is not nef, advancing understanding in the structure of such pairs.

Findings

Identifies conditions for non-nef adjoint bundles in corank 3 cases

Provides structural classification of pairs (X, E)

Derives consequences in adjunction theory

Abstract

Let $\mathcal{E}$ be an ample vector bundle of rank $r\geq 2$ on a smooth complex projective variety $X$ of dimension $n$. The aim of this paper is to describe the structure of pairs $(X,\mathcal{E})$ as above whose adjoint bundles $K_X+\det\mathcal{E}$ are not nef for $r=n-3$. Furthermore, we give some immediate consequences of this result in adjunction theory.