Fano manifolds whose elementary contractions are smooth P1 -fibrations: a geometric characterization of flag varieties

The present paper provides a geometric characterization of complete flag varieties for semisimple algebraic groups. Namely, we show that if X is a Fano manifold whose elementary contractions are all P1 -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