On rank 2 vector bundles on Fano manifolds

TL;DR

This paper investigates rank two vector bundles on Fano manifolds with specific Betti numbers, analyzing their geometric cones, classifying certain bundles, and establishing stability results.

Contribution

It provides a complete classification of uniform rank two bundles and explores the relationship between cones and bundle decomposability on these Fano manifolds.

Findings

Classified all uniform rank two vector bundles on the given Fano manifolds.

Identified all $P^1$-bundles with a second $P^1$-bundle structure.

Proved stability of indecomposable Fano bundles with one exception.

Abstract

In this work we deal with vector bundles of rank two on a Fano manifold $X$ with $b_2=b_4=1$. We study the nef and pseudoeffective cones of the corresponding projectivizations and how these cones are related to the decomposability of the vector bundle. As consequences, we obtain the complete list of $\mathbb{P}^1$-bundles over $X$ that have a second $\mathbb{P}^1$-bundle structure, classify all the uniform rank two vector bundles on this class of Fano manifolds and show the stability of indecomposable Fano bundles (with one exception on $\mathbb{P}^2$).