A survey on the Campana-Peternell Conjecture

TL;DR

This survey reviews the progress on the Campana-Peternell Conjecture, which aims to classify Fano manifolds with nef tangent bundles, highlighting recent partial results and ongoing research efforts.

Contribution

It provides a comprehensive overview of the background, recent advances, and remaining challenges in the classification of Fano manifolds with nef tangent bundles.

Findings

Partial classifications achieved for specific cases

Identification of key geometric properties influencing the conjecture

Progress towards confirming the conjecture in certain dimensions

Abstract

In 1991 Campana and Peternell proposed, as a natural algebro-geometric extension of Mori's characterization of the projective space, the problem of classifying the complex projective Fano manifolds whose tangent bundle is nef, conjecturing that the only varieties satisfying these properties are rational homogeneous. In this paper we review some background material related to this problem, with special attention to the partial results recently obtained by the authors.