---
abstract: 'We give a new proof of the classification due to Peternell-Szurek-Wiśniewski of nef vector bundles on a projective space with the first Chern class less than three and on a smooth hyperquadric with the first Chern class less than two over an algebraically closed field of characteristic zero.'
address: 'Graduate School of Informatics and Engineering, The University of Electro-Communications, Chofu-shi, Tokyo, 182-8585 Japan '
author:
- Masahiro Ohno
bibliography:
- 'NefOnHyperquadrics.bib'
title: Nef vector bundles on a projective space or a hyperquadric with the first Chern class small
---

Introduction
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Let $X$ be a smooth complex projective variety of dimension $n$, and $\mathcal{E}$ a nef vector bundle of rank $r$ on it. Many authors have studied a pair $(X,\mathcal{E})$ of $X$ and $\mathcal{E}$ in connection with classifications of special types of Fano (or Fano-like) manifolds, where $\mathcal{E}$ may be assumed to be spanned (i.e., globally generated) or ample, as the case may be. In these classifications, often appears a pair $(X, \mathbb{E})$ of $X$ of Picard number one and an ample vector bundle $\mathbb{E}$ with the adjoint bundle $K_X+\det\mathbb{E}$ trivial (i.e., isomorphic to the structure sheaf). In case $K_X+\det\mathbb{E}$ is trivial, we have $r\leqq n+1$ by Mori’s 