Fano manifolds whose elementary contractions are smooth $\mathbb P^1$-fibrations: A geometric characterization of flag varieties

Gianluca Occhetta, Luis E. Solá Conde, Kiwamu Watanabe, Jarosław A. Wiśniewski

Published 2014-07-14, updated 2016-01-26Version 3

The present paper provides a geometric characterization of complete flag varieties for semisimple algebraic groups. Namely, if $X$ is a Fano manifold whose all elementary contractions are $\mathbb P^1$-fibrations then $X$ is isomorphic to the complete flag manifold $G/B$ where $G$ is a semi-simple Lie algebraic group and $B$ is a Borel subgroup of $G$.