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

Abstract

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 $G$.

30 pages, minor issues corrected. To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

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