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

TL;DR

This paper characterizes complete flag varieties among Fano manifolds by showing that if all elementary contractions are smooth $P^1$-fibrations, then the manifold is isomorphic to a flag variety $G/B$ for a semisimple algebraic group.

Contribution

It provides a geometric characterization of flag varieties based on the structure of their elementary contractions, linking Fano manifolds to classical algebraic groups.

Findings

Fano manifolds with all elementary contractions as $P^1$-fibrations are isomorphic to flag varieties.

Establishes a criterion to identify flag varieties via their contraction properties.

Connects geometric properties of Fano manifolds to algebraic group structures.

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 $G$.