Flag bundles on Fano manifolds

Abstract

As an application of a recent characterization of complete flag manifolds as Fano manifolds having only P 1 -bundles as elementary contractions, we consider here the case of a Fano manifold X of Picard number one supporting an unsplit family of rational curves whose subfamilies parametrizing curves through a fixed point are rational homogeneous, and we prove that X is homogeneous. In order to do this, we first study minimal sections on flag bundles over the projective line, and discuss how Grothendieck's theorem on principal bundles allows us to describe a flag bundle upon some special sections.

Characterizing the homogeneous variety

“…), by a co‐character of a Cartan subgroup of G , modulo the action of the Weyl group of G . Following [, Section 3], this information may be interpreted geometrically as follows: for every minimal parabolic subgroup properly containing B , the morphism is a ‐bundle. Denoting by its relative canonical divisor, the equivalence class of the co‐character defining is determined by the set of intersection numbers , where Γ denotes a minimal section of the flag bundle over (see [, Proposition 3.17]).…”

“…Following [, Section 3], this information may be interpreted geometrically as follows: for every minimal parabolic subgroup properly containing B , the morphism is a ‐bundle. Denoting by its relative canonical divisor, the equivalence class of the co‐character defining is determined by the set of intersection numbers , where Γ denotes a minimal section of the flag bundle over (see [, Proposition 3.17]). Each of these parabolic subgroups corresponds to a node i of the Dynkin diagram , therefore we may represent the ‐bundle by the tagged Dynkin diagram obtained by appending the intersection number to the node i (see [, Remark 3.18]).…”

“…Given now a curve ℓ of the family , we may consider the image of θ by: which defines an ‐principal bundle over ℓ. Following [, Section 3.3], this principal bundle is classified by a tagged Dynkin diagram of type B 3 , where are nonnegative integers.…”

“…Recently we have proved a similar characterization of long‐root rational homogeneous spaces with algebraic methods, using a characterization of complete flag manifolds in terms of rational curves (see [, Theorem A.1]). Under this approach one needs to assume that the variety , parametrizing curves of through a point x , is the expected one at every point .…”

“…Upon it, by means of a detailed description of the projective geometry of the VMRT, we construct a complete flag bundle of type B 3 , and show that supports as many independent ‐bundle structures as its Picard number. Then we use [, Theorem A.1] to claim that this variety is a complete flag manifold; being the target of a contraction of a rational homogeneous manifold, X will then be rational homogeneous as well; we can then conclude because, among these manifolds, F 4 (4) is determined by its VMRT at a point.…”