On rank 2 vector bundles on Fano manifolds

Roberto Muñoz, Gianluca Occhetta, Luis Solá Conde

Published 2011-04-08, updated 2012-12-21Version 4

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 $\mathbb{P}^2$).